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# Lesson 10: The Chain Rule (handout)

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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 (handout)

1. 1. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Notes . Sec on 2.5 The Chain Rule V63.0121.001: Calculus I Professor Ma hew Leingang New York University February 23, 2011 . . Notes Announcements Quiz 2 next week on §§1.5, 1.6, 2.1, 2.2 Midterm March 7 on all sec ons in class (covers all sec ons up to 2.5) . . Notes Objectives Given a compound expression, write it as a composi on of func ons. Understand and apply the Chain Rule for the deriva ve of a composi on of func ons. Understand and use Newtonian and Leibnizian nota ons for the Chain Rule. . . . 1.
2. 2. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Compositions Notes See Section 1.2 for review Deﬁni on If f and g are func ons, the composi on (f ◦ g)(x) = f(g(x)) means “do g ﬁrst, then f.” x g f◦g g(x) . f f(g(x)) Our goal for the day is to understand how the deriva ve of the composi on of two func ons depends on the deriva ves of the individual func ons. . . Notes Outline Heuris cs Analogy The Linear Case The chain rule Examples . . Notes Analogy Think about riding a bike. To go faster you can either: pedal faster change gears radius of front sprocket . The angular posi on (φ) of the back wheel depends on the posi on of the front sprocket (θ): R.. θ φ(θ) = r.. And so the angular speed of the back wheel depends on the radius of back sprocket deriva ve of this func on and the speed of the front sprocket. . Image credit: SpringSun . . 2.
3. 3. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Notes The Linear Case Ques on Let f(x) = mx + b and g(x) = m′ x + b′ . What can you say about the composi on? Answer f(g(x)) = m(m′ x + b′ ) + b = (mm′ )x + (mb′ + b) The composi on is also linear The slope of the composi on is the product of the slopes of the two func ons. The deriva ve is supposed to be a local lineariza on of a func on. So there should be an analog of this property in deriva ves. . . Notes The Nonlinear Case Let u = g(x) and y = f(u). Suppose x is changed by a small amount ∆x. Then ∆y f′ (y) ≈ =⇒ ∆y ≈ f′ (y)∆u ∆x and ∆u g′ (y) ≈ =⇒ ∆u ≈ g′ (u)∆x. ∆x So ∆y ∆y ≈ f′ (y)g′ (u)∆x =⇒ ≈ f′ (y)g′ (u) ∆x . . Notes Outline Heuris cs Analogy The Linear Case The chain rule Examples . . . 3.
4. 4. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Notes Theorem of the day: The chain rule Theorem Let f and g be func ons, with g diﬀeren able at x and f diﬀeren able at g(x). Then f ◦ g is diﬀeren able at x and (f ◦ g)′ (x) = f′ (g(x))g′ (x) In Leibnizian nota on, let y = f(u) and u = g(x). Then dy dy du = dx du dx . . Notes Observations Succinctly, the deriva ve of a composi on is the product of the deriva ves The only complica on is where these deriva ves are evaluated: at the same point the func ons are In Leibniz nota on, the Chain Rule looks like cancella on of . (fake) frac ons . Image credit: ooOJasonOoo . Notes Outline Heuris cs Analogy The Linear Case The chain rule Examples . . . 4.
5. 5. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Notes Example Example √ let h(x) = 3x2 + 1. Find h′ (x). Solu on √ 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 . . Notes Corollary Corollary (The Power Rule Combined with the Chain Rule) If n is any real number and u = g(x) is diﬀeren able, then d n du (u ) = nun−1 . dx dx . . Notes Does order matter? Example d d Find (sin 4x) and compare it to (4 sin x). dx dx Solu on . . . 5.
6. 6. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Example Notes (√ )2 x − 2 + 8 . Find f′ (x). 3 5 Let f(x) = Solu on . . Notes A metaphor Think about peeling an onion: (√ )2 3 f(x) = x5 −2 +8 5 √ 3 +8 . 2 (√ ) f′ (x) = 2 x5 − 2 + 8 1 5 − 2)−2/3 (5x4 ) 3 3 (x . Image credit: photobunny . Notes Combining techniques Example d ( 3 ) Find (x + 1)10 sin(4x2 − 7) dx Solu on . . . 6.
7. 7. . V63.0121.001: Calculus I . Sec on 2.5: The Chain Rule . February 23, 2011 Notes Related rates of change in the ocean Ques on The area of a circle, A = πr2 , changes as its radius changes. If the radius changes with respect to me, the change in area with respect to me is dA A. = 2πr dr dA dr B. = 2πr + dt dt dA dr C. = 2πr dt dt . D. not enough informa on . Image credit: Jim Frazier . Notes Summary The deriva ve of a composi on is the product of deriva ves In symbols: (f ◦ g)′ (x) = f′ (g(x))g′ (x) . . Notes . . . 7.