Barrons AP Calculus - David Bock

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 = 4x^3 − 5x + 7, find y ′(1) and y ′′(1).
SOLUTION:
Then y ′(1) = 12 · 1^2 − 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