Differentiating a Nested Function

Date: 05/21/2000 at 13:34:45
From: Bob Phinney
Subject: Differentiating a function with a known derivative

The problem is: If g'(8) = 5, Find d/dx g(x^3) at x = 2.

I know how to differentiate a function when I have g(x) or g'(x) as a 
formula, but having the answer with a number plugged in for x is 
really confusing me. Can you give me a formula for using the g'(8) = 5 
term to determine the g'(x)?

Thanks.

Date: 05/23/2000 at 13:45:49
From: Doctor Douglas
Subject: Re: Differentiating a function with a known derivative

Hi Bob,

Thanks for writing to Ask Dr. Math.

You're absolutely right - sometimes having too much information is 
confusing. In this case, the g'(8) information isn't extraneous, but 
we're not yet ready for it, so it's confusing right now.

The question asks for d/dx g(x^3) at x = 2. Let's just use the chain 
rule first for the derivative of a composition of two functions
h(x) = g(f(x)), and see where it leads us.

     d/dx h(x) = d/dx g(f(x))     g(f) = g(f), f(x) = x^3
               = dg/df df/dx      derivatives with respect to the
               = g'(f) df/dx           corresponding arguments
               = g'(f) 3x^2       substitute for f(x) and take df/dx

Note that at this point, we don't know what the functional form of g 
is. We need to hope that somehow the derivative at 8 is going to be 
useful. Now on to x = 2:

     d/dx h(x) |(x=2) = g'(f) 3x^2 |(x=2)     evaluated at x=2
                      = g'(f(x=2)) 3*2^2
                      = g'(2^3) 12            aha!
                      = 12 g'(8)              if we didn't know g'(8)
                                                we'd stop here
                      = 12*5
                      = 60

Hope that helps. Please write back if you have additional questions.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/