Section 2.5
                    The Chain Rule

                     V63.0121.021, Calculus I

                          N...
Announcements




         Quiz 2 in recitation next
         week (October 11-15)
         Midterm in class Tuesday,
    ...
Objectives


         Given a compound
         expression, write it as a
         composition of functions.
         Unde...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Outline



Heuristics
  Analogy
  The Linear Case


The chain rule


Examples


Related rates of change



               ...
Analogy



    Think about riding a bike. To go
    faster you can either:




                                           ...
Analogy



    Think about riding a bike. To go
    faster you can either:
         pedal faster



                      ...
Analogy



    Think about riding a bike. To go
    faster you can either:
         pedal faster
         change gears


 ...
Analogy



    Think about riding a bike. To go
    faster you can either:
         pedal faster
         change gears


 ...
Analogy



    Think about riding a bike. To go
    faster you can either:
         pedal faster
         change gears


 ...
Analogy



    Think about riding a bike. To go
    faster you can either:
         pedal faster
         change gears


 ...
The Linear Case


Question
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the
composition?




          ...
The Linear Case


Question
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the
composition?

Answer

     ...
The Linear Case


Question
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the
composition?

Answer

     ...
The Linear Case


Question
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the
composition?

Answer

     ...
The Linear Case


Question
Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the
composition?

Answer

     ...
The Nonlinear Case



Let u = g(x) and y = f(u). Suppose x is changed by a small amount
∆x. Then
                         ...
Outline



Heuristics
  Analogy
  The Linear Case


The chain rule


Examples


Related rates of change



               ...
Theorem of the day: The chain rule


Theorem
Let f and g be functions, with g differentiable at x and f differentiable at
...
Observations



       Succinctly, the derivative of a
       composition is the product of
       the derivatives




   ...
Theorem of the day: The chain rule


Theorem
Let f and g be functions, with g differentiable at x and f differentiable at
...
Observations



       Succinctly, the derivative of a
       composition is the product of
       the derivatives
       ...
Compositions
See Section 1.2 for review


Definition
If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means ...
Observations



       Succinctly, the derivative of a
       composition is the product of
       the derivatives
       ...
Theorem of the day: The chain rule


Theorem
Let f and g be functions, with g differentiable at x and f differentiable at
...
Theorem of the day: The chain rule


Theorem
Let f and g be functions, with g differentiable at x and f differentiable at
...
Outline



Heuristics
  Analogy
  The Linear Case


The chain rule


Examples


Related rates of change



               ...
Example


Example
               √
let h(x) =         3x2 + 1. Find h′ (x).




                                          ...
Example


Example
               √
let h(x) =         3x2 + 1. Find h′ (x).

Solution
First, write h as f ◦ g.




       ...
Example


Example
               √
let h(x) =         3x2 + 1. Find h′ (x).

Solution
                                    ...
Example


Example
               √
let h(x) =         3x2 + 1. Find h′ (x).

Solution
                                    ...
Example


Example
               √
let h(x) =         3x2 + 1. Find h′ (x).

Solution
                                    ...
Corollary




Corollary (The Power Rule Combined with the Chain Rule)
If n is any real number and u = g(x) is differentiab...
Does order matter?

Example
     d                             d
Find    (sin 4x) and compare it to    (4 sin x).
     dx ...
Does order matter?

Example
     d                             d
Find    (sin 4x) and compare it to    (4 sin x).
     dx ...
Does order matter?

Example
     d                             d
Find    (sin 4x) and compare it to    (4 sin x).
     dx ...
Order matters!

Example
     d                             d
Find    (sin 4x) and compare it to    (4 sin x).
     dx     ...
Example
                (√                  )2
                                         . Find f′ (x).
                   ...
Example
                (√                  )2
                                         . Find f′ (x).
                   ...
Example
                (√                  )2
                                         . Find f′ (x).
                   ...
Example
                (√                  )2
                                         . Find f′ (x).
                   ...
Example
                (√                  )2
                                         . Find f′ (x).
                   ...
Example
                (√                  )2
                                         . Find f′ (x).
                   ...
A metaphor


    Think about peeling an onion:
                   (√                         )2
                      3
  ...
Combining techniques

Example
     d ( 3                    )
Find    (x + 1)10 sin(4x2 − 7)
     dx




                 ...
Combining techniques

Example
     d ( 3                    )
Find    (x + 1)10 sin(4x2 − 7)
     dx

Solution
The “last” ...
Combining techniques

Example
     d ( 3                    )
Find    (x + 1)10 sin(4x2 − 7)
     dx

Solution
The “last” ...
Combining techniques

Example
     d ( 3                    )
Find    (x + 1)10 sin(4x2 − 7)
     dx

Solution
The “last” ...
Your Turn


Find derivatives of these functions:
 1. y = (1 − x2 )10
        √
 2. y = sin x
           √
 3. y = sin x
 4...
Solution to #1




Example
Find the derivative of y = (1 − x2 )10 .

Solution
y′ = 10(1 − x2 )9 (−2x) = −20x(1 − x2 )9



...
Solution to #2



Example
                                        √
Find the derivative of y =               sin x.

Solut...
Solution to #3



Example
                              √
Find the derivative of y = sin x.

Solution
                    ...
Solution to #4
Example
Find the derivative of y = (2x − 5)4 (8x2 − 5)−3

Solution
We need to use the product rule and the ...
Solution to #5


Example
                                         √
                                             z−1
Find ...
Solution to #6




Example
Find the derivative of y = tan(cos x).

Solution
y′ = sec2 (cos x) · (− sin x) = − sec2 (cos x)...
Solution to #7

Example
Find the derivative of y = csc2 (sin θ).

Solution
Remember the notation:

                       ...
Solution to #8

Example
Find the derivative of y = sin(sin(sin(sin(sin(sin(x)))))).

Solution
Relax! It’s just a bunch of ...
Outline



Heuristics
  Analogy
  The Linear Case


The chain rule


Examples


Related rates of change



               ...
Related rates of change at the Deli


Question
Suppose a deli clerk can slice a stick of pepperoni (assume the
tapered end...
Related rates of change at the Deli


Question
Suppose a deli clerk can slice a stick of pepperoni (assume the
tapered end...
Related rates of change in the ocean

    Question
    The area of a circle, A = πr2 ,
    changes as its radius changes.
...
Related rates of change in the ocean

    Question
    The area of a circle, A = πr2 ,
    changes as its radius changes.
...
Summary



      The derivative of a
      composition is the product
      of derivatives
      In symbols:
      (f ◦ g)...
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Lesson 10: The Chain Rule (Section 21 slides)

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The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.

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Lesson 10: The Chain Rule (Section 21 slides)

  1. 1. Section 2.5 The Chain Rule V63.0121.021, Calculus I New York University October 7, 2010 Announcements Quiz 2 in recitation next week (October 11-15) Midterm in class Tuesday, october 19 on §§1.1–2.5 . . . . . .
  2. 2. Announcements Quiz 2 in recitation next week (October 11-15) Midterm in class Tuesday, october 19 on §§1.1–2.5 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 2 / 36
  3. 3. Objectives Given a compound expression, write it as a composition of functions. Understand and apply the Chain Rule for the derivative of a composition of functions. Understand and use Newtonian and Leibnizian notations for the Chain Rule. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 3 / 36
  4. 4. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 4 / 36
  5. 5. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” x . g . (x) g . . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 4 / 36
  6. 6. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” x . g . (x) g . . f . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 4 / 36
  7. 7. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” x . g . (x) f .(g(x)) g . . f . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 4 / 36
  8. 8. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” g . (x) f .(g(x)) . ◦ g x . g . f . f . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 4 / 36
  9. 9. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” g . (x) f .(g(x)) . ◦ g x . g . f . f . Our goal for the day is to understand how the derivative of the composition of two functions depends on the derivatives of the individual functions. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 4 / 36
  10. 10. Outline Heuristics Analogy The Linear Case The chain rule Examples Related rates of change . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 5 / 36
  11. 11. Analogy Think about riding a bike. To go faster you can either: . . Image credit: SpringSun . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 6 / 36
  12. 12. Analogy Think about riding a bike. To go faster you can either: pedal faster . . Image credit: SpringSun . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 6 / 36
  13. 13. Analogy Think about riding a bike. To go faster you can either: pedal faster change gears . . Image credit: SpringSun . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 6 / 36
  14. 14. Analogy Think about riding a bike. To go faster you can either: pedal faster change gears . The angular position (φ) of the back wheel depends on the position of the front sprocket (θ): R.θ . φ(θ) = r. . . Image credit: SpringSun . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 6 / 36
  15. 15. Analogy Think about riding a bike. To go faster you can either: pedal faster change gears r . adius of front sprocket . The angular position (φ) of the back wheel depends on the position of the front sprocket (θ): R.θ . φ(θ) = r. . r . adius of back sprocket . Image credit: SpringSun . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 6 / 36
  16. 16. Analogy Think about riding a bike. To go faster you can either: pedal faster change gears . The angular position (φ) of the back wheel depends on the position of the front sprocket (θ): R.θ . φ(θ) = r. . And so the angular speed of the back wheel depends on the derivative of this function and the speed of the front sprocket. . Image credit: SpringSun . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 6 / 36
  17. 17. The Linear Case Question Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composition? . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 7 / 36
  18. 18. The Linear Case Question Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composition? Answer f(g(x)) = m(m′ x + b′ ) + b = (mm′ )x + (mb′ + b) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 7 / 36
  19. 19. The Linear Case Question Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composition? Answer f(g(x)) = m(m′ x + b′ ) + b = (mm′ )x + (mb′ + b) The composition is also linear . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 7 / 36
  20. 20. The Linear Case Question Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composition? Answer f(g(x)) = m(m′ x + b′ ) + b = (mm′ )x + (mb′ + b) The composition is also linear The slope of the composition is the product of the slopes of the two functions. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 7 / 36
  21. 21. The Linear Case Question Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composition? Answer f(g(x)) = m(m′ x + b′ ) + b = (mm′ )x + (mb′ + b) The composition is also linear The slope of the composition is the product of the slopes of the two functions. The derivative is supposed to be a local linearization of a function. So there should be an analog of this property in derivatives. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 7 / 36
  22. 22. The Nonlinear Case Let u = g(x) and y = f(u). Suppose x is changed by a small amount ∆x. Then ∆y ≈ f′ (y)∆u and ∆u ≈ g′ (u)∆x. So ∆y ∆y ≈ f′ (y)g′ (u)∆x =⇒ ≈ f′ (y)g′ (u) ∆x . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 8 / 36
  23. 23. Outline Heuristics Analogy The Linear Case The chain rule Examples Related rates of change . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 9 / 36
  24. 24. Theorem of the day: The chain rule Theorem Let f and g be functions, with g differentiable at x and f differentiable at g(x). Then f ◦ g is differentiable at x and (f ◦ g)′ (x) = f′ (g(x))g′ (x) In Leibnizian notation, let y = f(u) and u = g(x). Then dy dy du = dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 10 / 36
  25. 25. Observations Succinctly, the derivative of a composition is the product of the derivatives . . Image credit: ooOJasonOoo . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 11 / 36
  26. 26. Theorem of the day: The chain rule Theorem Let f and g be functions, with g differentiable at x and f differentiable at g(x). Then f ◦ g is differentiable at x and (f ◦ g)′ (x) = f′ (g(x))g′ (x) In Leibnizian notation, let y = f(u) and u = g(x). Then dy dy du = dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 12 / 36
  27. 27. Observations Succinctly, the derivative of a composition is the product of the derivatives The only complication is where these derivatives are evaluated: at the same point the functions are . . Image credit: ooOJasonOoo . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 13 / 36
  28. 28. Compositions See Section 1.2 for review Definition If f and g are functions, the composition (f ◦ g)(x) = f(g(x)) means “do g first, then f.” g . (x) f .(g(x)) . ◦ g x . g . f . f . . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 14 / 36
  29. 29. Observations Succinctly, the derivative of a composition is the product of the derivatives The only complication is where these derivatives are evaluated: at the same point the functions are In Leibniz notation, the Chain Rule looks like cancellation of (fake) fractions . . Image credit: ooOJasonOoo . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 15 / 36
  30. 30. Theorem of the day: The chain rule Theorem Let f and g be functions, with g differentiable at x and f differentiable at g(x). Then f ◦ g is differentiable at x and (f ◦ g)′ (x) = f′ (g(x))g′ (x) In Leibnizian notation, let y = f(u) and u = g(x). Then dy dy du = dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 16 / 36
  31. 31. Theorem of the day: The chain rule Theorem Let f and g be functions, with g differentiable at x and f differentiable at g(x). Then f ◦ g is differentiable at x and (f ◦ g)′ (x) = f′ (g(x))g′ (x) dy In Leibnizian notation, let y = f(u) and u = g(x).du Then . . dx du dy dy du = dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 16 / 36
  32. 32. Outline Heuristics Analogy The Linear Case The chain rule Examples Related rates of change . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 17 / 36
  33. 33. Example Example √ let h(x) = 3x2 + 1. Find h′ (x). . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 18 / 36
  34. 34. Example Example √ let h(x) = 3x2 + 1. Find h′ (x). Solution First, write h as f ◦ g. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 18 / 36
  35. 35. Example Example √ let h(x) = 3x2 + 1. Find h′ (x). Solution √ First, write h as f ◦ g. Let f(u) = u and g(x) = 3x2 + 1. . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 18 / 36
  36. 36. Example Example √ let h(x) = 3x2 + 1. Find h′ (x). Solution √ First, write h as f ◦ g. Let f(u) = u and g(x) = 3x2 + 1. Then f′ (u) = 1 u−1/2 , and g′ (x) = 6x. So 2 h′ (x) = 1 u−1/2 (6x) 2 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 18 / 36
  37. 37. Example Example √ let h(x) = 3x2 + 1. Find h′ (x). Solution √ First, write h as f ◦ g. Let f(u) = u and g(x) = 3x2 + 1. Then f′ (u) = 1 u−1/2 , and g′ (x) = 6x. So 2 3x h′ (x) = 1 u−1/2 (6x) = 1 (3x2 + 1)−1/2 (6x) = √ 2 2 3x2 + 1 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 18 / 36
  38. 38. Corollary Corollary (The Power Rule Combined with the Chain Rule) If n is any real number and u = g(x) is differentiable, then d n du (u ) = nun−1 . dx dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 19 / 36
  39. 39. Does order matter? Example d d Find (sin 4x) and compare it to (4 sin x). dx dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 20 / 36
  40. 40. Does order matter? Example d d Find (sin 4x) and compare it to (4 sin x). dx dx Solution For the first, let u = 4x and y = sin(u). Then dy dy du = · = cos(u) · 4 = 4 cos 4x. dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 20 / 36
  41. 41. Does order matter? Example d d Find (sin 4x) and compare it to (4 sin x). dx dx Solution For the first, let u = 4x and y = sin(u). Then dy dy du = · = cos(u) · 4 = 4 cos 4x. dx du dx For the second, let u = sin x and y = 4u. Then dy dy du = · = 4 · cos x dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 20 / 36
  42. 42. Order matters! Example d d Find (sin 4x) and compare it to (4 sin x). dx dx Solution For the first, let u = 4x and y = sin(u). Then dy dy du = · = cos(u) · 4 = 4 cos 4x. dx du dx For the second, let u = sin x and y = 4u. Then dy dy du = · = 4 · cos x dx du dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 20 / 36
  43. 43. Example (√ )2 . Find f′ (x). 3 Let f(x) = x5 − 2 + 8 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 21 / 36
  44. 44. Example (√ )2 . Find f′ (x). 3 Let f(x) = x5 − 2 + 8 Solution d (√ 5 3 )2 (√ 3 ) d (√ 3 ) x −2+8 =2 x5 − 2 + 8 x5 − 2 + 8 dx dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 21 / 36
  45. 45. Example (√ )2 . Find f′ (x). 3 Let f(x) = x5 − 2 + 8 Solution d (√ 5 3 )2 (√ 3 ) d (√ 3 ) x −2+8 =2 x5 − 2 + 8 x5 − 2 + 8 dx dx (√ ) d√ 3 3 =2 x5 − 2 + 8 x5 − 2 dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 21 / 36
  46. 46. Example (√ )2 . Find f′ (x). 3 Let f(x) = x5 − 2 + 8 Solution d (√ 5 3 )2 (√ 3 ) d (√ 3 ) x −2+8 =2 x5 − 2 + 8 x5 − 2 + 8 dx dx (√ ) d√ 3 3 =2 x5 − 2 + 8 x5 − 2 dx (√ ) d x5 − 2 + 8 1 (x5 − 2)−2/3 (x5 − 2) 3 =2 3 dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 21 / 36
  47. 47. Example (√ )2 . Find f′ (x). 3 Let f(x) = x5 − 2 + 8 Solution d (√ 5 3 )2 (√ 3 ) d (√ 3 ) x −2+8 =2 x5 − 2 + 8 x5 − 2 + 8 dx dx (√ ) d√ 3 3 =2 x5 − 2 + 8 x5 − 2 dx (√ ) d x5 − 2 + 8 1 (x5 − 2)−2/3 (x5 − 2) 3 =2 3 (√ ) dx =2 3 x 5 − 2 + 8 1 (x5 − 2)−2/3 (5x4 ) 3 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 21 / 36
  48. 48. Example (√ )2 . Find f′ (x). 3 Let f(x) = x5 − 2 + 8 Solution d (√ 5 3 )2 (√ 3 ) d (√ 3 ) x −2+8 =2 x5 − 2 + 8 x5 − 2 + 8 dx dx (√ ) d√ 3 3 =2 x5 − 2 + 8 x5 − 2 dx (√ ) d x5 − 2 + 8 1 (x5 − 2)−2/3 (x5 − 2) 3 =2 3 (√ ) dx =2 3 x 5 − 2 + 8 1 (x5 − 2)−2/3 (5x4 ) 3 (√ 10 4 3 5 ) = x x − 2 + 8 (x5 − 2)−2/3 3 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 21 / 36
  49. 49. A metaphor Think about peeling an onion: (√ )2 3 f(x) = x 5 −2 +8 5 √ 3 +8 . (√ ) 2 f′ (x) = 2 x5 − 2 + 8 1 (x5 − 2)−2/3 (5x4 ) 3 3 . Image credit: photobunny . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 22 / 36
  50. 50. Combining techniques Example d ( 3 ) Find (x + 1)10 sin(4x2 − 7) dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 23 / 36
  51. 51. Combining techniques Example d ( 3 ) Find (x + 1)10 sin(4x2 − 7) dx Solution The “last” part of the function is the product, so we apply the product rule. Each factor’s derivative requires the chain rule: . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 23 / 36
  52. 52. Combining techniques Example d ( 3 ) Find (x + 1)10 sin(4x2 − 7) dx Solution The “last” part of the function is the product, so we apply the product rule. Each factor’s derivative requires the chain rule: d ( 3 ) (x + 1)10 · sin(4x2 − 7) dx ( ) ( ) d 3 d = (x + 1) 10 · sin(4x − 7) + (x + 1) · 2 3 10 sin(4x − 7) 2 dx dx . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 23 / 36
  53. 53. Combining techniques Example d ( 3 ) Find (x + 1)10 sin(4x2 − 7) dx Solution The “last” part of the function is the product, so we apply the product rule. Each factor’s derivative requires the chain rule: d ( 3 ) (x + 1)10 · sin(4x2 − 7) dx ( ) ( ) d 3 d = (x + 1) 10 · sin(4x − 7) + (x + 1) · 2 3 10 sin(4x − 7) 2 dx dx = 10(x3 + 1)9 (3x2 ) sin(4x2 − 7) + (x3 + 1)10 · cos(4x2 − 7)(8x) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 23 / 36
  54. 54. Your Turn Find derivatives of these functions: 1. y = (1 − x2 )10 √ 2. y = sin x √ 3. y = sin x 4. y = (2x − 5)4 (8x2 − 5)−3 √ z−1 5. F(z) = z+1 6. y = tan(cos x) 7. y = csc2 (sin θ) 8. y = sin(sin(sin(sin(sin(sin(x)))))) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 24 / 36
  55. 55. Solution to #1 Example Find the derivative of y = (1 − x2 )10 . Solution y′ = 10(1 − x2 )9 (−2x) = −20x(1 − x2 )9 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 25 / 36
  56. 56. Solution to #2 Example √ Find the derivative of y = sin x. Solution √ Writing sin x as (sin x)1/2 , we have cos x y′ = 1 2 (sin x)−1/2 (cos x) = √ 2 sin x . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 26 / 36
  57. 57. Solution to #3 Example √ Find the derivative of y = sin x. Solution (√ ) ′ d 1/2 1 −1/2 cos x y = 1/2 sin(x ) = cos(x ) 2 x = √ dx 2 x . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 27 / 36
  58. 58. Solution to #4 Example Find the derivative of y = (2x − 5)4 (8x2 − 5)−3 Solution We need to use the product rule and the chain rule: y′ = 4(2x − 5)3 (2)(8x2 − 5)−3 + (2x − 5)4 (−3)(8x2 − 5)−4 (16x) The rest is a bit of algebra, useful if you wanted to solve the equation y′ = 0: [ ] y′ = 8(2x − 5)3 (8x2 − 5)−4 (8x2 − 5) − 6x(2x − 5) ( ) = 8(2x − 5)3 (8x2 − 5)−4 −4x2 + 30x − 5 ( ) = −8(2x − 5)3 (8x2 − 5)−4 4x2 − 30x + 5 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 28 / 36
  59. 59. Solution to #5 Example √ z−1 Find the derivative of F(z) = . z+1 Solution ( )−1/2 ( ) 1 z−1(z + 1)(1) − (z − 1)(1) y′ = 2 z+1 (z + 1)2 ( )1/2 ( ) 1 z+1 2 1 = = 2 z−1 (z + 1)2 (z + 1)3/2 (z − 1)1/2 . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 29 / 36
  60. 60. Solution to #6 Example Find the derivative of y = tan(cos x). Solution y′ = sec2 (cos x) · (− sin x) = − sec2 (cos x) sin x . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 30 / 36
  61. 61. Solution to #7 Example Find the derivative of y = csc2 (sin θ). Solution Remember the notation: y = csc2 (sin θ) = [csc(sin θ)]2 So y′ = 2 csc(sin θ) · [− csc(sin θ) cot(sin θ)] · cos(θ) = −2 csc2 (sin θ) cot(sin θ) cos θ . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 31 / 36
  62. 62. Solution to #8 Example Find the derivative of y = sin(sin(sin(sin(sin(sin(x)))))). Solution Relax! It’s just a bunch of chain rules. All of these lines are multiplied together. y′ = cos(sin(sin(sin(sin(sin(x)))))) · cos(sin(sin(sin(sin(x))))) · cos(sin(sin(sin(x)))) · cos(sin(sin(x))) · cos(sin(x)) · cos(x)) . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 32 / 36
  63. 63. Outline Heuristics Analogy The Linear Case The chain rule Examples Related rates of change . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 33 / 36
  64. 64. Related rates of change at the Deli Question Suppose a deli clerk can slice a stick of pepperoni (assume the tapered ends have been removed) by hand at the rate of 2 inches per minute, while a machine can slice pepperoni at the rate of 10 inches dV dV per minute. Then for the machine is 5 times greater than for dt dt the deli clerk. This is explained by the A. chain rule B. product rule C. quotient Rule D. addition rule . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 34 / 36
  65. 65. Related rates of change at the Deli Question Suppose a deli clerk can slice a stick of pepperoni (assume the tapered ends have been removed) by hand at the rate of 2 inches per minute, while a machine can slice pepperoni at the rate of 10 inches dV dV per minute. Then for the machine is 5 times greater than for dt dt the deli clerk. This is explained by the A. chain rule B. product rule C. quotient Rule D. addition rule . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 34 / 36
  66. 66. Related rates of change in the ocean Question The area of a circle, A = πr2 , changes as its radius changes. If the radius changes with respect to time, the change in area with respect to time is dA A. = 2πr dr dA dr B. = 2πr + dt dt . dA dr C. = 2πr dt dt D. not enough information . Image credit: Jim Frazier . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 35 / 36
  67. 67. Related rates of change in the ocean Question The area of a circle, A = πr2 , changes as its radius changes. If the radius changes with respect to time, the change in area with respect to time is dA A. = 2πr dr dA dr B. = 2πr + dt dt . dA dr C. = 2πr dt dt D. not enough information . Image credit: Jim Frazier . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 35 / 36
  68. 68. Summary The derivative of a composition is the product of derivatives In symbols: (f ◦ g)′ (x) = f′ (g(x))g′ (x) Calculus is like an onion, and not because it makes you cry! . . . . . . V63.0121.021, Calculus I (NYU) Section 2.5 The Chain Rule October 7, 2010 36 / 36

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