Lesson 8: Basic Differentiation Rules (Section 41 handout)

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Lesson 8: Basic Differentiation Rules (Section 41 handout)

  1. 1. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Notes Section 2.3 Basic Differentiation Rules V63.0121.041, Calculus I New York University September 28, 2010 Announcements Last chance for extra credit on Quiz 1: Do the get-to-know you survey and photo by October 1. Announcements Notes Last chance for extra credit on Quiz 1: Do the get-to-know you survey and photo by October 1. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 2 / 42 Objectives Notes Understand and use these differentiation rules: the derivative of a constant function (zero); the Constant Multiple Rule; the Sum Rule; the Difference Rule; the derivatives of sine and cosine. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 3 / 42 1
  2. 2. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Recall: the derivative Notes 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. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 4 / 42 Notation Notes Newtonian notation Leibnizian notation dy d df f (x) y (x) y f (x) dx dx dx V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 5 / 42 Link between the notations Notes 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 quotients, dx not as an actual fraction itself. The notation suggests things which are true even though they don’t follow from the notation per se V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 6 / 42 2
  3. 3. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Outline Notes 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 7 / 42 Derivative of the squaring function Notes Example Suppose f (x) = x 2 . Use the definition of derivative to find f (x). Solution f (x + h) − f (x) (x + h)2 − x 2 f (x) = lim = lim h→0 h h→0 h x2 2 x2   + 2xh + h −       2 2x h + h¡ ¡ = lim = lim h→0 h h→0 h ¡ = lim (2x + h) = 2x. h→0 So f (x) = 2x. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 8 / 42 The second derivative Notes 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: d 2y d2 d 2f f (x) dx 2 dx 2 dx 2 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 9 / 42 3
  4. 4. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 The squaring function and its derivatives Notes y f f f increasing =⇒ f ≥ 0 f f decreasing =⇒ f ≤ 0 x horizontal tangent at 0 =⇒ f (0) = 0 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 10 / 42 Derivative of the cubing function Notes Example Suppose f (x) = x 3 . Use the definition of derivative to find f (x). Solution V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 11 / 42 The cubing function and its derivatives Notes y Notice that f is increasing, f f and f > 0 except f (0) = 0 Notice also that the tangent f line to the graph of f at x (0, 0) crosses the graph (contrary to a popular “definition” of the tangent line) V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 12 / 42 4
  5. 5. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Derivative of the square root function Notes Example √ Suppose f (x) = x = x 1/2 . 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 13 / 42 The square root function and its derivatives Notes y f Here lim+ f (x) = ∞ and f x→0 f is not differentiable at 0 x Notice also lim f (x) = 0 x→∞ V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 14 / 42 Derivative of the cube root function Notes Example √ Suppose f (x) = 3 x = x 1/3 . Use the definition of derivative to find f (x). Solution V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 15 / 42 5
  6. 6. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 The cube root function and its derivatives Notes y Here lim f (x) = ∞ and f is f x→0 not differentiable at 0 f x Notice also lim f (x) = 0 x→±∞ V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 16 / 42 One more Notes Example Suppose f (x) = x 2/3 . Use the definition of derivative to find f (x). Solution V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 17 / 42 The function x → x 2/3 and its derivative Notes y f is not differentiable at 0 f and lim f (x) = ±∞ x→0± f x Notice also lim f (x) = 0 x→±∞ V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 18 / 42 6
  7. 7. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Recap: The Tower of Power Notes y y x2 2x 1 The power goes down by x3 3x 2 one in each derivative 1 −1/2 The coefficient in the x 1/2 2x derivative is the power of 1 −2/3 x 1/3 3x the original function 2 −1/3 x 2/3 3x V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 19 / 42 The Power Rule Notes There is mounting evidence for Theorem (The Power Rule) Let r be a real number and f (x) = x r . Then f (x) = rx r −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 . V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 20 / 42 The other Tower of Power Notes V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 21 / 42 7
  8. 8. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Outline Notes 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 22 / 42 Remember your algebra Notes Fact Let n be a positive whole number. Then (x + h)n = x n + nx n−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) = ck x k hn−k n copies k=0 n The coefficient of x is 1 because we have to choose x from each binomial, and there’s only one way to do that. The coefficient of x n−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. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 23 / 42 Pascal’s Triangle Notes 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 = 1x 2 + 2xh + 1h2 1 5 10 10 5 1 (x + h)3 = 1x 3 + 3x 2 h + 3xh2 + 1h3 ... ... 1 6 15 20 15 6 1 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 24 / 42 8
  9. 9. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Proving the Power Rule Notes Theorem (The Power Rule) Let r be a positive whole number. Then d r x = rx r −1 dx Proof. As we showed above, (x + h)n = x n + nx n−1 h + (stuff with at least two hs in it) So (x + h)n − x n nx n−1 h + (stuff with at least two hs in it) = h h = nx n−1 + (stuff with at least one h in it) and this tends to nx n−1 as h → 0. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 25 / 42 The Power Rule for constants Notes Theorem d 0 like x = 0x −1 Let c be a constant. Then 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 26 / 42 Calculus Notes 9
  10. 10. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Recall the Limit Laws Notes 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. . . . V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 28 / 42 Adding functions Notes 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 . V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 29 / 42 Proof of the Sum Rule Notes 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. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 30 / 42 10
  11. 11. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Scaling functions Notes 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 . V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 31 / 42 Proof of the Constant Multiple Rule Notes 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) V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 32 / 42 Derivatives of polynomials Notes Example d Find 2x 3 + x 4 − 17x 12 + 37 dx Solution d 2x 3 + x 4 − 17x 12 + 37 dx d d 4 d d = 2x 3 + x + −17x 12 + (37) dx dx dx dx d d 4 d = 2 x3 + x − 17 x 12 + 0 dx dx dx = 2 · 3x 2 + 4x 3 − 17 · 12x 11 = 6x 2 + 4x 3 − 204x 11 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 33 / 42 11
  12. 12. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Outline Notes 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 34 / 42 Derivatives of Sine and Cosine Notes 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 35 / 42 Angle addition formulas See Appendix A Notes V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 36 / 42 12
  13. 13. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 Two important trigonometric limits See Section 1.4 Notes sin θ lim =1 θ→0 θ sin θ θ cos θ − 1 lim =0 θ θ→0 θ 1 − cos θ 1 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 38 / 42 Illustration of Sine and Cosine Notes y x π −π 0 π π 2 2 cos x sin x f (x) = sin x has horizontal tangents where f = cos(x) is zero. what happens at the horizontal tangents of cos? V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 40 / 42 Derivatives of Sine and Cosine Notes 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 V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 41 / 42 13
  14. 14. V63.0121.041, Calculus I Section 2.3 : Basic Differentiation Rules September 28, 2010 What have we learned today? Notes 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. V63.0121.041, Calculus I (NYU) Section 2.3 Basic Differentiation Rules September 28, 2010 42 / 42 Notes Notes 14

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