Doug C. answered • 10/03/23
Math Tutor with Reputation to make difficult concepts understandable
Assume that the functions f and g are both functions of x. That means when we take the derivative with respect to x of g (for example), the chain rule must be applied, i.e. multiply by the derivative of the "inside".
For [f(x) g(x)2]' the product rule needs to be applied. I usually think, first times derivative of 2nd + 2nd times derivative of the first--I know a lot of textbooks these days teach it as f'g + fg'.
For convenience represent this as: h(x ) = f(x) g(x)2. Then,
h'() = f(x) 2g(x)g'(x) + g(x)2f'(x) [product rule, where f(x) is the first function and g(x)2 is the 2nd]
You might want to think of g(x)2 as [g(x)]2. Then apply the power rule followed by the chain rule: 2[g(x)]1g'(x).
Rearranged:
h'(x) = 2f(x)g(x)g'(x) + f'(x) g(x)2
h'(3) = 2f(3)g(3)g'(3) + f'(3) g(3)2
Substitute the given values and evaluate.
