Section	2.3
         Basic	Differentiation	Rules

               V63.0121.006/016, Calculus	I



                     Febr...
Recall: the	derivative

   Definition
   Let f be	a	function	and a a	point	in	the	domain	of f. If	the	limit

              ...
Notation

      Newtonian	notation       Leibnizian	notation
                               dy       d             df
    ...
Link	between	the	notations



                         f(x + ∆x) − f(x)       ∆y   dy
          f′ (x) = lim              ...
Outline


  Derivatives	so	far
     Derivatives	of	power	functions	by	hand
     The	Power	Rule


  Derivatives	of	polynomi...
Derivative	of	the	squaring	function

   Example
   Suppose f(x) = x2 . Use	the	definition	of	derivative	to	find f′ (x).




...
Derivative	of	the	squaring	function

   Example
   Suppose f(x) = x2 . Use	the	definition	of	derivative	to	find f′ (x).

   ...
Derivative	of	the	squaring	function

   Example
   Suppose f(x) = x2 . Use	the	definition	of	derivative	to	find f′ (x).

   ...
Derivative	of	the	squaring	function

   Example
   Suppose f(x) = x2 . Use	the	definition	of	derivative	to	find f′ (x).

   ...
Derivative	of	the	squaring	function

   Example
   Suppose f(x) = x2 . Use	the	definition	of	derivative	to	find f′ (x).

   ...
Derivative	of	the	squaring	function

   Example
   Suppose f(x) = x2 . Use	the	definition	of	derivative	to	find f′ (x).

   ...
The	second	derivative



   If f is	a	function, so	is f′ , and	we	can	seek	its	derivative.

                              ...
The	second	derivative



   If f is	a	function, so	is f′ , and	we	can	seek	its	derivative.

                              ...
The	squaring	function	and	its	derivatives



           y
           .




           .        x
                    .



...
The	squaring	function	and	its	derivatives



           y
           .




           .   f
               .
             ...
The	squaring	function	and	its	derivatives



           y
           .
               .′
               f
                ...
The	squaring	function	and	its	derivatives



           y
           .
               .′
               f
                ...
Derivative	of	the	cubing	function

   Example
   Suppose f(x) = x3 . Use	the	definition	of	derivative	to	find f′ (x).




  ...
Derivative	of	the	cubing	function

   Example
   Suppose f(x) = x3 . Use	the	definition	of	derivative	to	find f′ (x).

   So...
Derivative	of	the	cubing	function

   Example
   Suppose f(x) = x3 . Use	the	definition	of	derivative	to	find f′ (x).

   So...
Derivative	of	the	cubing	function

   Example
   Suppose f(x) = x3 . Use	the	definition	of	derivative	to	find f′ (x).

   So...
Derivative	of	the	cubing	function

   Example
   Suppose f(x) = x3 . Use	the	definition	of	derivative	to	find f′ (x).

   So...
The	cubing	function	and	its	derivatives




                  y
                  .




                  .           x
  ...
The	cubing	function	and	its	derivatives




                  y
                  .



                       f
          ...
The	cubing	function	and	its	derivatives



                                   Notice	that f is
                  y
       ...
The	cubing	function	and	its	derivatives



                                           Notice	that f is
                  y...
The	cubing	function	and	its	derivatives



                                           Notice	that f is
                  y...
Derivative	of	the	square	root	function
   Example
                    √
   Suppose f(x) =    x = x1/2 . Use	the	definition	...
Derivative	of	the	square	root	function
   Example
                    √
   Suppose f(x) =    x = x1/2 . Use	the	definition	...
Derivative	of	the	square	root	function
   Example
                    √
   Suppose f(x) =    x = x1/2 . Use	the	definition	...
Derivative	of	the	square	root	function
   Example
                    √
   Suppose f(x) =    x = x1/2 . Use	the	definition	...
Derivative	of	the	square	root	function
   Example
                    √
   Suppose f(x) =    x = x1/2 . Use	the	definition	...
Derivative	of	the	square	root	function
   Example
                    √
   Suppose f(x) =    x = x1/2 . Use	the	definition	...
The	square	root	function	and	its	derivatives



    y
    .




     .              x
                    .




          ...
The	square	root	function	and	its	derivatives



    y
    .


         f
         .
     .              x
                ...
The	square	root	function	and	its	derivatives



    y
    .

                                    Here lim f′ (x) = ∞ and
 ...
The	square	root	function	and	its	derivatives



    y
    .

                                    Here lim f′ (x) = ∞ and
 ...
Derivative	of	the	cube	root	function
   Example
                    √
   Suppose f(x) =   3
                      x = x1/3...
Derivative	of	the	cube	root	function
   Example
                    √
   Suppose f(x) =   3
                      x = x1/3...
Derivative	of	the	cube	root	function
   Example
                    √
   Suppose f(x) =   3
                      x = x1/3...
Derivative	of	the	cube	root	function
   Example
                    √
   Suppose f(x) =   3
                      x = x1/3...
Derivative	of	the	cube	root	function
   Example
                     √
   Suppose f(x) =    3
                       x = x...
Derivative	of	the	cube	root	function
   Example
                      √
   Suppose f(x) =     3
                        x ...
The	cube	root	function	and	its	derivatives




                y
                .




                 .              x
 ...
The	cube	root	function	and	its	derivatives




                y
                .


                     f
              ...
The	cube	root	function	and	its	derivatives




                y
                .
                                       ...
The	cube	root	function	and	its	derivatives




                y
                .
                                       ...
One	more

  Example
  Suppose f(x) = x2/3 . Use	the	definition	of	derivative	to	find f′ (x).




                           ...
One	more

  Example
  Suppose f(x) = x2/3 . Use	the	definition	of	derivative	to	find f′ (x).

  Solution

                  ...
One	more

  Example
  Suppose f(x) = x2/3 . Use	the	definition	of	derivative	to	find f′ (x).

  Solution

                  ...
One	more

  Example
  Suppose f(x) = x2/3 . Use	the	definition	of	derivative	to	find f′ (x).

  Solution

                  ...
One	more

  Example
  Suppose f(x) = x2/3 . Use	the	definition	of	derivative	to	find f′ (x).

  Solution

                  ...
The	function x → x2/3 and	its	derivative




                y
                .




                 .              x
   ...
The	function x → x2/3 and	its	derivative




                y
                .


                     f
                ...
The	function x → x2/3 and	its	derivative




                y
                .
                                         ...
The	function x → x2/3 and	its	derivative




                y
                .
                                         ...
Recap



         y       y′
        x2      2x
        x3      3x2
               1 −1/2
        x1/2   2x
              ...
Recap



         y       y′
        x2      2x
        x3      3x2
               1 −1/2
        x1/2   2x
              ...
Recap



         y       y′
        x2      2x
        x3      3x2
               1 −1/2
        x1/2   2x
              ...
Recap



         y       y′
        x2      2x
        x3      3x2
               1 −1/2
        x1/2   2x
              ...
Recap



         y       y′
        x2      2x
        x3      3x2
               1 −1/2
        x1/2   2x
              ...
Recap



         y       y′
        x2      2x1
        x3      3x2
               1 −1/2
        x1/2   2x
             ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x1          The	power	goes	down
                          ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x           The	power	goes	down
                          ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x           The	power	goes	down
                          ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x           The	power	goes	down
                          ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x           The	power	goes	down
                          ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x           The	power	goes	down
                          ...
Recap: The	Tower	of	Power



        y       y′
        x2     2x           The	power	goes	down
                          ...
The	Power	Rule


  There	is	mounting	evidence	for
  Theorem	(The	Power	Rule)
  Let r be	a	real	number	and f(x) = xr . Then...
The	other	Tower	of	Power




                           .   .   .   .   .   .
Outline


  Derivatives	so	far
     Derivatives	of	power	functions	by	hand
     The	Power	Rule


  Derivatives	of	polynomi...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Remember	your	algebra
  Fact
  Let n be	a	positive	whole	number. Then

         (x + h)n = xn + nxn−1 h + (stuff	with	at	l...
Pascal’s	Triangle


                           ..
                           1

                       1
                 ...
Pascal’s	Triangle


                           ..
                           1

                       1
                 ...
Pascal’s	Triangle


                           ..
                           1

                       1
                 ...
Pascal’s	Triangle


                           ..
                           1

                       1
                 ...
Theorem	(The	Power	Rule)
Let r be	a	positive	whole	number. Then

                         d r
                            ...
Theorem	(The	Power	Rule)
Let r be	a	positive	whole	number. Then

                            d r
                         ...
The	Power	Rule	for	constants

   Theorem
   Let c be	a	constant. Then
                               d
                   ...
The	Power	Rule	for	constants

   Theorem
                                                    d 0
   Let c be	a	constant. T...
The	Power	Rule	for	constants

   Theorem
                                                    d 0
   Let c be	a	constant. T...
New	derivatives	from	old
This	is	where	the	calculus	starts	to	get	really	powerful!




                                   ...
Calculus



           .   .   .   .   .   .
Recall	the	Limit	Laws




   Fact
   Suppose lim f(x) = L and lim g(x) = M and c is	a	constant. Then
                 x→a ...
Adding	functions



   Theorem	(The	Sum	Rule)
   Let f and g be	functions	and	define

                           (f + g)(x)...
Proof.
Follow	your	nose:

                         (f + g)(x + h) − (f + g)(x)
     (f + g)′ (x) = lim
                   ...
Proof.
Follow	your	nose:

                       (f + g)(x + h) − (f + g)(x)
     (f + g)′ (x) = lim
                  h →...
Proof.
Follow	your	nose:

                       (f + g)(x + h) − (f + g)(x)
     (f + g)′ (x) = lim
                  h →...
Proof.
Follow	your	nose:

                        (f + g)(x + h) − (f + g)(x)
      (f + g)′ (x) = lim
                   ...
Scaling	functions



   Theorem	(The	Constant	Multiple	Rule)
   Let f be	a	function	and c a	constant. Define

             ...
Proof.
Again, follow	your	nose.

                                (cf)(x + h) − (cf)(x)
               (cf)′ (x) = lim
    ...
Proof.
Again, follow	your	nose.

                              (cf)(x + h) − (cf)(x)
               (cf)′ (x) = lim
      ...
Proof.
Again, follow	your	nose.

                              (cf)(x + h) − (cf)(x)
               (cf)′ (x) = lim
      ...
Proof.
Again, follow	your	nose.

                                (cf)(x + h) − (cf)(x)
               (cf)′ (x) = lim
    ...
Derivatives	of	polynomials

   Example
          d ( 3                   )
   Find      2x + x4 − 17x12 + 37
          dx
...
Derivatives	of	polynomials

   Example
          d ( 3                   )
   Find      2x + x4 − 17x12 + 37
          dx
...
Derivatives	of	polynomials

   Example
          d ( 3                   )
   Find      2x + x4 − 17x12 + 37
          dx
...
Derivatives	of	polynomials

   Example
          d ( 3                   )
   Find      2x + x4 − 17x12 + 37
          dx
...
Derivatives	of	polynomials

   Example
          d ( 3                   )
   Find      2x + x4 − 17x12 + 37
          dx
...
Outline


  Derivatives	so	far
     Derivatives	of	power	functions	by	hand
     The	Power	Rule


  Derivatives	of	polynomi...
Derivatives	of	Sine	and	Cosine
   Fact
                         d
                            sin x = ???
                ...
Derivatives	of	Sine	and	Cosine
   Fact
                              d
                                 sin x = ???
      ...
Angle	addition	formulas
See	Appendix	A




                 .
                                      .
                    ...
Derivatives	of	Sine	and	Cosine
   Fact
                               d
                                  sin x = ???
    ...
Derivatives	of	Sine	and	Cosine
   Fact
                                d
                                   sin x = ???
  ...
Two	important	trigonometric	limits
See	Section	1.4




                                           .
                      ...
Derivatives	of	Sine	and	Cosine
   Fact
                                d
                                   sin x = ???
  ...
Derivatives	of	Sine	and	Cosine
   Fact
                                d
                                   sin x = ???
  ...
Derivatives	of	Sine	and	Cosine
   Fact
                              d
                                 sin x = cos x
    ...
Illustration	of	Sine	and	Cosine



                          y
                          .


                            ....
Illustration	of	Sine	and	Cosine



                               y
                               .


                   ...
Illustration	of	Sine	and	Cosine



                               y
                               .


                   ...
Derivatives	of	Sine	and	Cosine
   Fact
             d                  d
                sin x = cos x      cos x = − sin ...
Derivatives	of	Sine	and	Cosine
   Fact
                 d                   d
                    sin x = cos x       cos ...
Derivatives	of	Sine	and	Cosine
   Fact
                  d                    d
                     sin x = cos x        ...
Derivatives	of	Sine	and	Cosine
   Fact
                  d                     d
                     sin x = cos x       ...
Derivatives	of	Sine	and	Cosine
   Fact
                  d                     d
                     sin x = cos x       ...
What	have	we	learned	today?




      The	Power	Rule




                              .   .   .   .   .   .
What	have	we	learned	today?




      The	Power	Rule
      The	derivative	of	a	sum	is	the	sum	of	the	derivatives
      The...
What	have	we	learned	today?




      The	Power	Rule
      The	derivative	of	a	sum	is	the	sum	of	the	derivatives
      The...
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Lesson 8: Basic Differentiation Rules

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The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.

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Lesson 8: Basic Differentiation Rules

  1. 1. Section 2.3 Basic Differentiation Rules V63.0121.006/016, Calculus I February 11, 2010 Announcements Quiz tomorrow in recitation. . . . . . .
  2. 2. Recall: the derivative Definition Let f be a function and a a point in the domain of f. If the limit f(a + h) − f(a) f(x) − f(a) f′ (a) = lim = lim h →0 h x →a x−a exists, the function is said to be differentiable at a and f′ (a) is the derivative of f at a. The derivative … …measures the slope of the line through (a, f(a)) tangent to the curve y = f(x); …represents the instantaneous rate of change of f at a …produces the best possible linear approximation to f near a. . . . . . .
  3. 3. Notation Newtonian notation Leibnizian notation dy d df f′ (x) y ′ (x ) y′ f(x) dx dx dx . . . . . .
  4. 4. Link between the notations f(x + ∆x) − f(x) ∆y dy f′ (x) = lim = lim = ∆x→0 ∆x ∆x→0 ∆x dx dy Leibniz thought of as a quotient of “infinitesimals” dx dy We think of as representing a limit of (finite) difference dx quotients, not as an actual fraction itself. The notation suggests things which are true even though they don’t follow from the notation per se . . . . . .
  5. 5. Outline Derivatives so far Derivatives of power functions by hand The Power Rule Derivatives of polynomials The Power Rule for whole number powers The Power Rule for constants The Sum Rule The Constant Multiple Rule Derivatives of sine and cosine . . . . . .
  6. 6. Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (x). . . . . . .
  7. 7. Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) f′ (x) = lim h →0 h . . . . . .
  8. 8. Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2 − x2 f′ (x) = lim = lim h →0 h h →0 h . . . . . .
  9. 9. Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2 − x2 f′ (x) = lim = lim h →0 h h →0 h 2 x2 x2   + 2xh + h −       = lim h →0 h . . . . . .
  10. 10. Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2 − x2 f′ (x) = lim = lim h →0 h h →0 h 2 2 x2 x2   + 2xh + h −       2xh + h¡ ¡ = lim = lim h →0 h h →0 h ¡ . . . . . .
  11. 11. Derivative of the squaring function Example Suppose f(x) = x2 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2 − x2 f′ (x) = lim = lim h →0 h h →0 h 2 2 x2 x2   + 2xh + h −       2xh + h¡ ¡ = lim = lim h →0 h h →0 h ¡ = lim (2x + h) = 2x. h →0 So f′ (x) = 2x. . . . . . .
  12. 12. The second derivative If f is a function, so is f′ , and we can seek its derivative. f′′ = (f′ )′ It measures the rate of change of the rate of change! . . . . . .
  13. 13. The second derivative If f is a function, so is f′ , and we can seek its derivative. f′′ = (f′ )′ It measures the rate of change of the rate of change! Leibnizian notation: d2 y d2 d2 f f(x) dx2 dx2 dx2 . . . . . .
  14. 14. The squaring function and its derivatives y . . x . . . . . . .
  15. 15. The squaring function and its derivatives y . . f . x . . . . . . .
  16. 16. The squaring function and its derivatives y . .′ f f increasing =⇒ f′ ≥ 0 f . f decreasing =⇒ f′ ≤ 0 . x . horizontal tangent at 0 =⇒ f′ (0) = 0 . . . . . .
  17. 17. The squaring function and its derivatives y . .′ f .′′ f f increasing =⇒ f′ ≥ 0 f . f decreasing =⇒ f′ ≤ 0 . x . horizontal tangent at 0 =⇒ f′ (0) = 0 . . . . . .
  18. 18. Derivative of the cubing function Example Suppose f(x) = x3 . Use the definition of derivative to find f′ (x). . . . . . .
  19. 19. Derivative of the cubing function Example Suppose f(x) = x3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)3 − x3 f′ (x) = lim = lim h →0 h h →0 h . . . . . .
  20. 20. Derivative of the cubing function Example Suppose f(x) = x3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)3 − x3 f′ (x) = lim = lim h →0 h h →0 h 2 3 x3 2 x3   + 3x h + 3xh + h −       = lim h →0 h . . . . . .
  21. 21. Derivative of the cubing function Example Suppose f(x) = x3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)3 − x3 f′ (x) = lim = lim h →0 h h →0 h 1 2 x3 2 2 3 x3 3x2 h ¡ ! 2 ! ¡ 3   + 3x h + 3xh + h −       ¡ + 3xh + h = lim = lim h →0 h h →0 h ¡ . . . . . .
  22. 22. Derivative of the cubing function Example Suppose f(x) = x3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)3 − x3 f′ (x) = lim = lim h →0 h h →0 h 1 2 x3 2 2 3 x3 3x2 h ¡ ! 2 ! ¡ 3   + 3x h + 3xh + h −       ¡ + 3xh + h = lim = lim h →0 h h →0 h ¡ ( ) = lim 3x2 + 3xh + h2 = 3x2 . h →0 So f′ (x) = 3x2 . . . . . . .
  23. 23. The cubing function and its derivatives y . . x . . . . . . .
  24. 24. The cubing function and its derivatives y . f . . x . . . . . . .
  25. 25. The cubing function and its derivatives Notice that f is y . f . ′ increasing, and f′ > 0 except f′ (0) = 0 f . . x . . . . . . .
  26. 26. The cubing function and its derivatives Notice that f is y . f . ′′ f . ′ increasing, and f′ > 0 except f′ (0) = 0 f . . x . . . . . . .
  27. 27. The cubing function and its derivatives Notice that f is y . f . ′′ f . ′ increasing, and f′ > 0 except f′ (0) = 0 Notice also that the f . tangent line to the graph . x . of f at (0, 0) crosses the graph (contrary to a popular “definition” of the tangent line) . . . . . .
  28. 28. Derivative of the square root function Example √ Suppose f(x) = x = x1/2 . Use the definition of derivative to find f′ (x). . . . . . .
  29. 29. Derivative of the square root function Example √ Suppose f(x) = x = x1/2 . Use the definition of derivative to find f′ (x). Solution √ √ ′ f(x + h) − f(x) x+h− x f (x) = lim = lim h →0 h h →0 h . . . . . .
  30. 30. Derivative of the square root function Example √ Suppose f(x) = x = x1/2 . Use the definition of derivative to find f′ (x). Solution √ √ ′ f(x + h) − f(x) x+h− x f (x) = lim = lim h →0 h h →0 h √ √ √ √ x+h− x x+h+ x = lim ·√ √ h →0 h x+h+ x . . . . . .
  31. 31. Derivative of the square root function Example √ Suppose f(x) = x = x1/2 . Use the definition of derivative to find f′ (x). Solution √ √ ′ f(x + h) − f(x) x+h− x f (x) = lim = lim h →0 h h →0 h √ √ √ √ x+h− x x+h+ x = lim ·√ √ h →0 h x+h+ x (¡ + h) − ¡ x x = lim (√ √ ) h →0 h x+h+ x . . . . . .
  32. 32. Derivative of the square root function Example √ Suppose f(x) = x = x1/2 . Use the definition of derivative to find f′ (x). Solution √ √ ′ f(x + h) − f(x) x+h− x f (x) = lim = lim h →0 h h →0 h √ √ √ √ x+h− x x+h+ x = lim ·√ √ h →0 h x+h+ x (¡ + h) − ¡ x x h ¡ = lim (√ √ ) = lim (√ √ ) h →0 h x+h+ x h →0 h ¡ x+h+ x . . . . . .
  33. 33. Derivative of the square root function Example √ Suppose f(x) = x = x1/2 . Use the definition of derivative to find f′ (x). Solution √ √ ′ f(x + h) − f(x) x+h− x f (x) = lim = lim h →0 h h →0 h √ √ √ √ x+h− x x+h+ x = lim ·√ √ h →0 h x+h+ x (¡ + h) − ¡ x x h ¡ = lim (√ √ ) = lim (√ √ ) h →0 h x+h+ x h →0 h ¡ x+h+ x 1 = √ 2 x √ So f′ (x) = x = 1 x−1/2 . 2 . . . . . .
  34. 34. The square root function and its derivatives y . . x . . . . . . .
  35. 35. The square root function and its derivatives y . f . . x . . . . . . .
  36. 36. The square root function and its derivatives y . Here lim f′ (x) = ∞ and f . x →0 + f is not differentiable at 0 . .′ f x . . . . . . .
  37. 37. The square root function and its derivatives y . Here lim f′ (x) = ∞ and f . x →0 + f is not differentiable at 0 . .′ f x . Notice also lim f′ (x) = 0 x→∞ . . . . . .
  38. 38. Derivative of the cube root function Example √ Suppose f(x) = 3 x = x1/3 . Use the definition of derivative to find f′ (x). . . . . . .
  39. 39. Derivative of the cube root function Example √ Suppose f(x) = 3 x = x1/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)1/3 − x1/3 f′ (x) = lim = lim h →0 h h →0 h . . . . . .
  40. 40. Derivative of the cube root function Example √ Suppose f(x) = 3 x = x1/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)1/3 − x1/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 = lim · h →0 h (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 . . . . . .
  41. 41. Derivative of the cube root function Example √ Suppose f(x) = 3 x = x1/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)1/3 − x1/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 = lim · h →0 h (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 (¡ + h) − ¡ x x = lim ( 2/3 + (x + h)1/3 x1/3 + x2/3 ) h→0 h (x + h) . . . . . .
  42. 42. Derivative of the cube root function Example √ Suppose f(x) = 3 x = x1/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)1/3 − x1/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 = lim · h →0 h (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 (¡ + h) − ¡ x x = lim ( 2/3 + (x + h)1/3 x1/3 + x2/3 ) h→0 h (x + h) h ¡ = lim ( ) h →0 h (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 ¡ . . . . . .
  43. 43. Derivative of the cube root function Example √ Suppose f(x) = 3 x = x1/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)1/3 − x1/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 = lim · h →0 h (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 (¡ + h) − ¡ x x = lim ( 2/3 + (x + h)1/3 x1/3 + x2/3 ) h→0 h (x + h) h ¡ 1 = lim ( )= h →0 h (x + h)2/3 + (x + h)1/3 x1/3 + x2/3 ¡ 3x2/3 So f′ (x) = 1 x−2/3 . 3 . . . . . .
  44. 44. The cube root function and its derivatives y . . x . . . . . . .
  45. 45. The cube root function and its derivatives y . f . . x . . . . . . .
  46. 46. The cube root function and its derivatives y . Here lim f′ (x) = ∞ and f x →0 f . is not differentiable at 0 . .′ f x . . . . . . .
  47. 47. The cube root function and its derivatives y . Here lim f′ (x) = ∞ and f x →0 f . is not differentiable at 0 . .′ f Notice also x . lim f′ (x) = 0 x→±∞ . . . . . .
  48. 48. One more Example Suppose f(x) = x2/3 . Use the definition of derivative to find f′ (x). . . . . . .
  49. 49. One more Example Suppose f(x) = x2/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2/3 − x2/3 f′ (x) = lim = lim h →0 h h →0 h . . . . . .
  50. 50. One more Example Suppose f(x) = x2/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2/3 − x2/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 ( ) = lim · (x + h)1/3 + x1/3 h →0 h . . . . . .
  51. 51. One more Example Suppose f(x) = x2/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2/3 − x2/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 ( ) = lim · (x + h)1/3 + x1/3 h →0 h ( ) 1 −2/3 1 /3 = 3x 2x . . . . . .
  52. 52. One more Example Suppose f(x) = x2/3 . Use the definition of derivative to find f′ (x). Solution f(x + h) − f(x) (x + h)2/3 − x2/3 f′ (x) = lim = lim h →0 h h →0 h (x + h) 1/3 − x1/3 ( ) = lim · (x + h)1/3 + x1/3 h →0 h ( ) 1 −2/3 = 3x 2x 1 /3 = 2 x−1/3 3 So f′ (x) = 2 x−1/3 . 3 . . . . . .
  53. 53. The function x → x2/3 and its derivative y . . x . . . . . . .
  54. 54. The function x → x2/3 and its derivative y . f . . x . . . . . . .
  55. 55. The function x → x2/3 and its derivative y . f is not differentiable at f . 0 and lim f′ (x) = ±∞ x →0 ± ′ . f . x . . . . . . .
  56. 56. The function x → x2/3 and its derivative y . f is not differentiable at f . 0 and lim f′ (x) = ±∞ x →0 ± ′ . f . Notice also x . lim f′ (x) = 0 x→±∞ . . . . . .
  57. 57. Recap y y′ x2 2x x3 3x2 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  58. 58. Recap y y′ x2 2x x3 3x2 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  59. 59. Recap y y′ x2 2x x3 3x2 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  60. 60. Recap y y′ x2 2x x3 3x2 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  61. 61. Recap y y′ x2 2x x3 3x2 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  62. 62. Recap y y′ x2 2x1 x3 3x2 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  63. 63. Recap: The Tower of Power y y′ x2 2x1 The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  64. 64. Recap: The Tower of Power y y′ x2 2x The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  65. 65. Recap: The Tower of Power y y′ x2 2x The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  66. 66. Recap: The Tower of Power y y′ x2 2x The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  67. 67. Recap: The Tower of Power y y′ x2 2x The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  68. 68. Recap: The Tower of Power y y′ x2 2x The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x 1 −2/3 x1/3 3x 2 −1/3 x2/3 3x . . . . . .
  69. 69. Recap: The Tower of Power y y′ x2 2x The power goes down by one in each x3 3x2 derivative 1 −1/2 x1/2 2x The coefficient in the 1 −2/3 derivative is the power x1/3 3x of the original function 2 −1/3 x2/3 3x . . . . . .
  70. 70. The Power Rule There is mounting evidence for Theorem (The Power Rule) Let r be a real number and f(x) = xr . Then f′ (x) = rxr−1 as long as the expression on the right-hand side is defined. Perhaps the most famous rule in calculus We will assume it as of today We will prove it many ways for many different r. . . . . . .
  71. 71. The other Tower of Power . . . . . .
  72. 72. Outline Derivatives so far Derivatives of power functions by hand The Power Rule Derivatives of polynomials The Power Rule for whole number powers The Power Rule for constants The Sum Rule The Constant Multiple Rule Derivatives of sine and cosine . . . . . .
  73. 73. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) . . . . . .
  74. 74. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) Proof. We have n ∑ (x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = c k x k h n −k n copies k=0 . . . . . .
  75. 75. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) Proof. We have n ∑ (x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = c k x k h n −k n copies k=0 The coefficient of xn is 1 because we have to choose x from each binomial, and there’s only one way to do that. . . . . . .
  76. 76. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) Proof. We have n ∑ (x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = c k x k h n −k n copies k=0 The coefficient of xn is 1 because we have to choose x from each binomial, and there’s only one way to do that. The coefficient of xn−1 h is the number of ways we can choose x n − 1 times, which is the same as the number of different hs we can pick, which is n. . . . . . .
  77. 77. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) Proof. We have n ∑ (x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = c k x k h n −k n copies k=0 The coefficient of xn is 1 because we have to choose x from each binomial, and there’s only one way to do that. The coefficient of xn−1 h is the number of ways we can choose x n − 1 times, which is the same as the number of different hs we can pick, which is n. . . . . . .
  78. 78. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) Proof. We have n ∑ (x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = c k x k h n −k n copies k=0 The coefficient of xn is 1 because we have to choose x from each binomial, and there’s only one way to do that. The coefficient of xn−1 h is the number of ways we can choose x n − 1 times, which is the same as the number of different hs we can pick, which is n. . . . . . .
  79. 79. Remember your algebra Fact Let n be a positive whole number. Then (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) Proof. We have n ∑ (x + h)n = (x + h) · (x + h) · (x + h) · · · (x + h) = c k x k h n −k n copies k=0 The coefficient of xn is 1 because we have to choose x from each binomial, and there’s only one way to do that. The coefficient of xn−1 h is the number of ways we can choose x n − 1 times, which is the same as the number of different hs we can pick, which is n. . . . . . .
  80. 80. Pascal’s Triangle .. 1 1 . 1 . 1 . 2 . 1 . 1 . 3 . 3 . 1 . (x + h)0 = 1 1 . 4 . 6 . 4 . 1 . (x + h)1 = 1x + 1h (x + h)2 = 1x2 + 2xh + 1h2 1 . 5 1 1 5 . .0 .0 . 1 . (x + h)3 = 1x3 + 3x2 h + 3xh2 + 1h3 1 . 6 1 2 1 6 . .5 .0 .5 . 1 . ... ... . . . . . .
  81. 81. Pascal’s Triangle .. 1 1 . 1 . 1 . 2 . 1 . 1 . 3 . 3 . 1 . (x + h)0 = 1 1 . 4 . 6 . 4 . 1 . (x + h)1 = 1x + 1h (x + h)2 = 1x2 + 2xh + 1h2 1 . 5 1 1 5 . .0 .0 . 1 . (x + h)3 = 1x3 + 3x2 h + 3xh2 + 1h3 1 . 6 1 2 1 6 . .5 .0 .5 . 1 . ... ... . . . . . .
  82. 82. Pascal’s Triangle .. 1 1 . 1 . 1 . 2 . 1 . 1 . 3 . 3 . 1 . (x + h)0 = 1 1 . 4 . 6 . 4 . 1 . (x + h)1 = 1x + 1h (x + h)2 = 1x2 + 2xh + 1h2 1 . 5 1 1 5 . .0 .0 . 1 . (x + h)3 = 1x3 + 3x2 h + 3xh2 + 1h3 1 . 6 1 2 1 6 . .5 .0 .5 . 1 . ... ... . . . . . .
  83. 83. Pascal’s Triangle .. 1 1 . 1 . 1 . 2 . 1 . 1 . 3 . 3 . 1 . (x + h)0 = 1 1 . 4 . 6 . 4 . 1 . (x + h)1 = 1x + 1h (x + h)2 = 1x2 + 2xh + 1h2 1 . 5 1 1 5 . .0 .0 . 1 . (x + h)3 = 1x3 + 3x2 h + 3xh2 + 1h3 1 . 6 1 2 1 6 . .5 .0 .5 . 1 . ... ... . . . . . .
  84. 84. Theorem (The Power Rule) Let r be a positive whole number. Then d r x = rxr−1 dx . . . . . .
  85. 85. Theorem (The Power Rule) Let r be a positive whole number. Then d r x = rxr−1 dx Proof. As we showed above, (x + h)n = xn + nxn−1 h + (stuff with at least two hs in it) So (x + h)n − xn nxn−1 h + (stuff with at least two hs in it) = h h = nxn−1 + (stuff with at least one h in it) and this tends to nxn−1 as h → 0. . . . . . .
  86. 86. The Power Rule for constants Theorem Let c be a constant. Then d c=0 dx . . . . . .
  87. 87. The Power Rule for constants Theorem d 0 Let c be a constant. Then l .ike x = 0x−1 dx d c=0. dx (although x → 0x−1 is not defined at zero.) . . . . . .
  88. 88. The Power Rule for constants Theorem d 0 Let c be a constant. Then l .ike x = 0x−1 dx d c=0. dx (although x → 0x−1 is not defined at zero.) Proof. Let f(x) = c. Then f(x + h) − f(x) c−c = =0 h h So f′ (x) = lim 0 = 0. h →0 . . . . . .
  89. 89. New derivatives from old This is where the calculus starts to get really powerful! . . . . . .
  90. 90. Calculus . . . . . .
  91. 91. Recall the Limit Laws Fact Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then x→a x →a 1. lim [f(x) + g(x)] = L + M x →a 2. lim [f(x) − g(x)] = L − M x →a 3. lim [cf(x)] = cL x →a 4. . . . . . . . . .
  92. 92. Adding functions Theorem (The Sum Rule) Let f and g be functions and define (f + g)(x) = f(x) + g(x) Then if f and g are differentiable at x, then so is f + g and (f + g)′ (x) = f′ (x) + g′ (x). Succinctly, (f + g)′ = f′ + g′ . . . . . . .
  93. 93. Proof. Follow your nose: (f + g)(x + h) − (f + g)(x) (f + g)′ (x) = lim h →0 h . . . . . .
  94. 94. Proof. Follow your nose: (f + g)(x + h) − (f + g)(x) (f + g)′ (x) = lim h →0 h f(x + h) + g(x + h) − [f(x) + g(x)] = lim h →0 h . . . . . .
  95. 95. Proof. Follow your nose: (f + g)(x + h) − (f + g)(x) (f + g)′ (x) = lim h →0 h f(x + h) + g(x + h) − [f(x) + g(x)] = lim h →0 h f(x + h) − f(x) g(x + h) − g(x) = lim + lim h →0 h h →0 h . . . . . .
  96. 96. Proof. Follow your nose: (f + g)(x + h) − (f + g)(x) (f + g)′ (x) = lim h →0 h f(x + h) + g(x + h) − [f(x) + g(x)] = lim h →0 h f(x + h) − f(x) g(x + h) − g(x) = lim + lim h →0 h h →0 h ′ ′ = f (x) + g (x) Note the use of the Sum Rule for limits. Since the limits of the difference quotients for for f and g exist, the limit of the sum is the sum of the limits. . . . . . .
  97. 97. Scaling functions Theorem (The Constant Multiple Rule) Let f be a function and c a constant. Define (cf)(x) = cf(x) Then if f is differentiable at x, so is cf and (cf)′ (x) = c · f′ (x) Succinctly, (cf)′ = cf′ . . . . . . .
  98. 98. Proof. Again, follow your nose. (cf)(x + h) − (cf)(x) (cf)′ (x) = lim h →0 h . . . . . .
  99. 99. Proof. Again, follow your nose. (cf)(x + h) − (cf)(x) (cf)′ (x) = lim h →0 h cf(x + h) − cf(x) = lim h →0 h . . . . . .
  100. 100. Proof. Again, follow your nose. (cf)(x + h) − (cf)(x) (cf)′ (x) = lim h →0 h cf(x + h) − cf(x) = lim h →0 h f(x + h) − f(x) = c lim h→0 h . . . . . .
  101. 101. Proof. Again, follow your nose. (cf)(x + h) − (cf)(x) (cf)′ (x) = lim h →0 h cf(x + h) − cf(x) = lim h →0 h f(x + h) − f(x) = c lim h→0 h ′ = c · f (x) . . . . . .
  102. 102. Derivatives of polynomials Example d ( 3 ) Find 2x + x4 − 17x12 + 37 dx . . . . . .
  103. 103. Derivatives of polynomials Example d ( 3 ) Find 2x + x4 − 17x12 + 37 dx Solution d ( 3 ) 2x + x4 − 17x12 + 37 dx d ( 3) d d ( ) d = 2x + x4 + −17x12 + (37) dx dx dx dx . . . . . .
  104. 104. Derivatives of polynomials Example d ( 3 ) Find 2x + x4 − 17x12 + 37 dx Solution d ( 3 ) 2x + x4 − 17x12 + 37 dx d ( 3) d d ( ) d = 2x + x4 + −17x12 + (37) dx dx dx dx d d d = 2 x3 + x4 − 17 x12 + 0 dx dx dx . . . . . .
  105. 105. Derivatives of polynomials Example d ( 3 ) Find 2x + x4 − 17x12 + 37 dx Solution d ( 3 ) 2x + x4 − 17x12 + 37 dx d ( 3) d d ( ) d = 2x + x4 + −17x12 + (37) dx dx dx dx d d d = 2 x3 + x4 − 17 x12 + 0 dx dx dx = 2 · 3x2 + 4x3 − 17 · 12x11 . . . . . .
  106. 106. Derivatives of polynomials Example d ( 3 ) Find 2x + x4 − 17x12 + 37 dx Solution d ( 3 ) 2x + x4 − 17x12 + 37 dx d ( 3) d d ( ) d = 2x + x4 + −17x12 + (37) dx dx dx dx d d d = 2 x3 + x4 − 17 x12 + 0 dx dx dx = 2 · 3x2 + 4x3 − 17 · 12x11 = 6x2 + 4x3 − 204x11 . . . . . .
  107. 107. Outline Derivatives so far Derivatives of power functions by hand The Power Rule Derivatives of polynomials The Power Rule for whole number powers The Power Rule for constants The Sum Rule The Constant Multiple Rule Derivatives of sine and cosine . . . . . .
  108. 108. Derivatives of Sine and Cosine Fact d sin x = ??? dx Proof. From the definition: . . . . . .
  109. 109. Derivatives of Sine and Cosine Fact d sin x = ??? dx Proof. From the definition: d sin(x + h) − sin x sin x = lim dx h→0 h . . . . . .
  110. 110. Angle addition formulas See Appendix A . . sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B − sin A sin B . . . . . .
  111. 111. Derivatives of Sine and Cosine Fact d sin x = ??? dx Proof. From the definition: d sin(x + h) − sin x sin x = lim dx h→0 h ( sin x cos h + cos x sin h) − sin x = lim h→0 h . . . . . .
  112. 112. Derivatives of Sine and Cosine Fact d sin x = ??? dx Proof. From the definition: d sin(x + h) − sin x sin x = lim dx h→0 h ( sin x cos h + cos x sin h) − sin x = lim h→0 h cos h − 1 sin h = sin x · lim + cos x · lim h →0 h h →0 h . . . . . .
  113. 113. Two important trigonometric limits See Section 1.4 . sin θ lim . =1 θ→0 θ s . in θ . cos θ − 1 θ lim =0 . θ θ→0 θ . . − cos θ 1 1 . . . . . . .
  114. 114. Derivatives of Sine and Cosine Fact d sin x = ??? dx Proof. From the definition: d sin(x + h) − sin x sin x = lim dx h→0 h ( sin x cos h + cos x sin h) − sin x = lim h→0 h cos h − 1 sin h = sin x · lim + cos x · lim h →0 h h →0 h = sin x · 0 + cos x · 1 . . . . . .
  115. 115. Derivatives of Sine and Cosine Fact d sin x = ??? dx Proof. From the definition: d sin(x + h) − sin x sin x = lim dx h→0 h ( sin x cos h + cos x sin h) − sin x = lim h→0 h cos h − 1 sin h = sin x · lim + cos x · lim h →0 h h →0 h = sin x · 0 + cos x · 1 . . . . . .
  116. 116. Derivatives of Sine and Cosine Fact d sin x = cos x dx Proof. From the definition: d sin(x + h) − sin x sin x = lim dx h→0 h ( sin x cos h + cos x sin h) − sin x = lim h→0 h cos h − 1 sin h = sin x · lim + cos x · lim h →0 h h →0 h = sin x · 0 + cos x · 1 = cos x . . . . . .
  117. 117. Illustration of Sine and Cosine y . . x . . π −2 . π 0 . .π . π 2 . in x s . . . . . .
  118. 118. Illustration of Sine and Cosine y . . x . . π −2 . π 0 . .π . π 2 . os x c . in x s f(x) = sin x has horizontal tangents where f′ = cos(x) is zero. . . . . . .
  119. 119. Illustration of Sine and Cosine y . . x . . π −2 . π 0 . .π . π 2 . os x c . in x s f(x) = sin x has horizontal tangents where f′ = cos(x) is zero. what happens at the horizontal tangents of cos? . . . . . .
  120. 120. Derivatives of Sine and Cosine Fact d d sin x = cos x cos x = − sin x dx dx . . . . . .
  121. 121. Derivatives of Sine and Cosine Fact d d sin x = cos x cos x = − sin x dx dx Proof. We already did the first. The second is similar (mutatis mutandis): d cos(x + h) − cos x cos x = lim dx h→0 h . . . . . .
  122. 122. Derivatives of Sine and Cosine Fact d d sin x = cos x cos x = − sin x dx dx Proof. We already did the first. The second is similar (mutatis mutandis): d cos(x + h) − cos x cos x = lim dx h→0 h (cos x cos h − sin x sin h) − cos x = lim h→0 h . . . . . .
  123. 123. Derivatives of Sine and Cosine Fact d d sin x = cos x cos x = − sin x dx dx Proof. We already did the first. The second is similar (mutatis mutandis): d cos(x + h) − cos x cos x = lim dx h→0 h (cos x cos h − sin x sin h) − cos x = lim h→0 h cos h − 1 sin h = cos x · lim − sin x · lim h →0 h h →0 h . . . . . .
  124. 124. Derivatives of Sine and Cosine Fact d d sin x = cos x cos x = − sin x dx dx Proof. We already did the first. The second is similar (mutatis mutandis): d cos(x + h) − cos x cos x = lim dx h→0 h (cos x cos h − sin x sin h) − cos x = lim h→0 h cos h − 1 sin h = cos x · lim − sin x · lim h →0 h h →0 h = cos x · 0 − sin x · 1 = − sin x . . . . . .
  125. 125. What have we learned today? The Power Rule . . . . . .
  126. 126. What have we learned today? The Power Rule The derivative of a sum is the sum of the derivatives The derivative of a constant multiple of a function is that constant multiple of the derivative . . . . . .
  127. 127. What have we learned today? The Power Rule The derivative of a sum is the sum of the derivatives The derivative of a constant multiple of a function is that constant multiple of the derivative The derivative of sine is cosine The derivative of cosine is the opposite of sine. . . . . . .

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