Find h'(6) when h(x)=f(x)/g(x)

Question

Suppose that f(x) and g(x) are differentiable functions such that f(6)=3, f'(6)=4, g(6)=8, and g'(6)=1. Find h'(6) when h(x)=f(x)/g(x).

Answer 1

Hello Carter,

If h(x) = f(x)/g(x), we obtain the derivative of h(x) at any value of x using the Quotient Rule:

h'(x) =[f '(x)g(x) - f(x)g'(x)]/(g(x)²)

(Here it is assumed that f '(x) and g'(x) exist, and that g(x)≠0.) You can use the information given in the problem to find the derivative of h at x = 6. That is,

h'(6) =[f '(6)g(6) - f(6)g'(6)]/(g(6)²)

h'(6) =[4·8 - 3·1]/(8²)

h'(6) = 29/64.

Hope that helps!

Willian