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/
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