Calculus AB and Calculus BC

CHAPTER 3 Differentiation

C. THE CHAIN RULE; THE DERIVATIVE OF A COMPOSITE FUNCTION

Formula (3) says that

This formula is an application of the Chain Rule. For example, if we use formula (3) to find the derivative of (x² − x + 2)⁴, we get

In this last equation, if we let y = (x² − x + 2)⁴ and let u = x² − x + 2, then y = u⁴. The preceding derivative now suggests one form of the Chain Rule:

as before. Formula (3) the previous page gives the general case where y = uⁿ and u is a differentiable function of x.

Now suppose we think of y as the composite function f (g(x)), where y = f (u) and u = g(x) are differentiable functions. Then

Chain rule

as we obtained above. The Chain Rule tells us how to differentiate the composite function: “Find the derivative of the ‘outside’ function first, then multiply by the derivative of the ‘inside’ one.”

For example:

Many of the formulas listed above in §B and most of the illustrative examples that follow use the Chain Rule. Often the chain rule is used more than once in finding a derivative.

Note that the algebraic simplifications that follow are included only for completeness.

EXAMPLE 1

If y = 4x³ − 5x + 7, find y ′(1) and y ″(1).

SOLUTION:

Then y ′(1) = 12 · 1² − 5 = 7 and y ″(1) = 24 · 1 = 24.

EXAMPLE 2

If f (x) = (3x + 2)⁵, find f ′(x).

SOLUTION: f ′(x) = 5(3x + 2)⁴ · 3 = 15(3x + 2)⁴.

EXAMPLE 3

SOLUTION:

EXAMPLE 4

SOLUTION:

EXAMPLE 5

If s(t) = (t² + 1)(1 − t)², find s ′(t).

SOLUTION:

EXAMPLE 6

If f (t) = e^2t sin 3t, find f ′(0).

SOLUTION:

Then, f ′(0) = 1(3 · 1 + 2 · 0) = 3.

EXAMPLE 7

SOLUTION:

Note that neither f (v) nor f ′(v) exists where the denominator equals zero, namely, where 1 − 2v² = 0 or where v equals

EXAMPLE 8

If x ≠ 0, find f ′(x).

SOLUTION:

EXAMPLE 9

If y = tan (2x² + 1), find y ′.

SOLUTION: y ′ = 4x sec² (2x² + 1).

EXAMPLE 10

If x = cos³ (1 − 3θ), find

SOLUTION:

EXAMPLE 11

If y = e^{(sin x) + 1}, find

SOLUTION:

EXAMPLE 12

If y = (x + 1)ln²(x + 1), find

SOLUTION:

EXAMPLE 13

If g(x) = (1 + sin² 3x)⁴, find

SOLUTION:

Then

EXAMPLE 14

If y = sin⁻¹ x + find y ′.

SOLUTION:

EXAMPLE 15

If u = ln find

SOLUTION: .

EXAMPLE 16

If s = e^−t(sin t − cos t), find s ′.

SOLUTION:

EXAMPLE 17

Let y = 2u³ − 4u² + 5u − 3 and u = x² − x. Find

SOLUTION:

EXAMPLE 18

If y = sin (ax + b), with a and b constants, find

SOLUTION: = [cos(ax + b)] · a = a cos(ax + b).

EXAMPLE 19

If f (x) = ae^kx (with a and k constants), find f ′ and f ″.

SOLUTION: f ′(x) = kae^kx and f ″ = k² ae^kx.

EXAMPLE 20

If y = ln (kx), where k is a constant, find

SOLUTION: We can use both formula (13), and the Chain Rule to get

Alternatively, we can rewrite the given function using a property of logarithms: ln (kx) = ln k + ln x. Then

EXAMPLE 21

Given f (u) = u² − u and u = g(x) = x³ − 5 and F(x) = f (g(x)), evaluate F ′(2).

SOLUTION: F ′(2) = f ′(g(2))g ′(2) = f ′(3) · (12) = 5 · 12 = 60.

Now, since g ′(x) = 3x², g ′(2) = 12, and since f ′(u) = 2u − 1, f ′(3) = 5. Of course, we get exactly the same answer as follows.