3-6

 

The Chain Rule

 

If f and g are both differentiable and F = fg is the composite function defined by F(x) = f(g(x)), then F is differentiable and  is given by the product:

(x) = g (x)) ((x))

 

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then:

 

Note: In using the chain rule, we work from the outside to the inside. We differentiate outer function f  [at the inner function g (x)] and then we multiply by the derivative of the inner function.

 

      f           (g(x))    =                 (g(x))             (x)

 

 (outer equation: evaluated at inner function: derivative of outer function: evaluated at   inner function: derivative of inner function)

 

 

 

Example 1:

 

 

 

The Power Rule combined with The Chain Rule:

 

If n is any real number and u = g(x) is differentiable, then:

 

 

Alternatively:

 

Example 2:

 

Find  if f (x) =

 

1.)    Rewrite f:  f (x) =

 

2.)   

 

3.)    =

 

 

Example 3: (Power rule, Quotient rule, Chain rule)

 

Find  if f (x) =

 

      =

 

      u =          v =

             

 

             =

 

           

 =