The chain rule is used to find the derivative of a function that is made up of the composite of two functions. 

 

Example:  let's let f(x) = x2 - 5, and g(x) = 3x + 4

Then f[g(x)] = f(3x + 4) = (3x +4)2 - 5

Let's give f[g(X)] a simpler name; let's call it h(x).
  So to find h'(x) means to find the derivative of a composite of two functions.

 Here's the rule:

For h(x) = f[g(x)]

h'(x) = f/[g(x)] · g'(x)

This is sometimes remembered by saying "The derivative of the outside times the derivative of the inside."

Let's get the ingredients of this rule for our example.  We need f'[g(x)] and g'(x)

f(x) = x2 - 5  implies that f'(x) = 2x, so f'[g(x)] = 2[g(x)]=2(3x + 4) = 6x + 8

g(x) = 3x + 4 implies that g'(x) = 3.  We're ready to put it all together now:

h'(x) = f/[g(x)] · g'(x) = (6x + 8)(3) = 18x + 24.

We can check this result by noting that

 f[g(x)] = (3x +4)2 - 5 = 9x2 + 24x + 16 - 5 = 9x2 + 24x + 11
      h(x) = 9x2 + 24x + 11 ,   so again we get that h'(x) = 18x + 24

Now, you may say to yourself that the method we used to check with was as easy or easier than using the chain rule, and for the composite function in our example, you would be right.  Actually, the chain rule is especially useful for a composite like:

h(x) = f[g(x)] = (4x2 + 5)3

where g(x) = 4x2 + 5   and   f(x) = x3

  f '(x) = 3x2, so that f '[g(x)] =3[g(x)]2 = 3(4x2 + 5)2 , and g'(x) = 8x  

                         Putting this altogether, we get, for
                                         h(x) =(4x2 + 5)3 

                     h'(x) = f/[g(x)] · g'(x) = 3(4x2 + 5)2 · 8x = 24x(4x2 + 5)2

 

Practice  Problem:   h(x) = (3x3 + 4x)5.      Find h'(x) using the chain rule  

(Work it out first; then scroll down for answer)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Practice  Problem Answer:   h'(x) = 5(3x3 + 4x)4(9x2 + 4)