Barrons AP Calculus

Chain   rule

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: and

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 __

If