**chain rule**provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. For example, if a composite function

*f*(

*x*) is defined as

Note that because two functions, *g* and *h*, make up the composite function *f*, you have to consider the derivatives *g*′ and *h*′ in differentiating *f*( *x*).

If a composite function *r*( *x*) is defined as

Here, three functions— *m*, *n*, and *p—*make up the composition function *r*; hence, you have to consider the derivatives *m′*, *n′*, and *p′* in differentiating *r*( *x*). A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken.

**Example 1:** Find *f′*( *x*) if *f*( *x*) = (3x ^{2} + 5x − 2) ^{8}.

**Example 2:** Find *f′*( *x*) if *f*( *x*) = tan (sec *x*).

**Example 3:** Find *y* = sin ^{3} (3 *x* − 1).

**Example 4:** Find *f*′(2) if

**Example 5:** Find the slope of the tangent line to a curve *y* = ( *x* ^{2} − 3) ^{5} at the point (−1, −32).

Because the slope of the tangent line to a curve is the derivative, you find that

which represents the slope of the tangent line at the point (−1,−32).