if f(3)=4, f'(3)=-4 g(3)=2, g'(3)=-6, find F'(3) if F(x)=cuberoot of (f(x)g(x))

Question

I am very confused as to how to find the derivative of (f(x)g(x))^(1/3)

Answer 1

The key is to realize the technique you will be using to find the derivative of F.

Since F(x) = [f(x)g(x)]^1/3 you will be using the power rule followed by the chain rule, i.e. derivative of the "inside"). But taking the derivative of the "inside" requires the use of the product rule.

So F'(x) = 1/3 [f(x)g(x)]^-2/3 [ f(x)g'(x} + g(x)f'(x}].

Since you are given the values for f(3), g(3) and f'(3), g'(3) you can substitute those values into the "formula" for F(x) and evaluate F'(3)--left to the student.

Answer 2

F(x) = [f(x)g(x)]^1/3

F'(x) = (1/3)[f(x)g(x)]^-2/3[f(x)g(x)]'

= (1/3)[f(x)g(x)]^-2/3[f'(x)g(x) + f(x)g'(x)]

So, F'(3) = (1/3)8^-2/3[(-4)(2) + (4)(-6)] = (1/12)[-8 - 24] = -8/3