Chain Rule

The 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  if 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) ⁵at 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).