Calculus Question

Question

Given that g(5) = −3, g'(5) = 7, h(5) = 3, and h'(5) = −4 find f '(5), if possible. (If not possible, enter IMPOSSIBLE.) f(x) = g(x)h(x)

Answer 1

The first thing you want to do is use the product rule to find f'(x). Since f(x)=g(x)h(x):

f'(x)=g'(x)h(x)+h'(x)g'(x)

Therefore:

f'(5)=g'(5)h(5)+h'(5)g(5)

Since we know the values for g and h and their derivatives at x=5, we can plug those in:

f'(5) = (7)(3)+(-4)(-3) = 21+12 = 33