Suppose h(x)=x^2 +2x−8h

Question

 g(3)=2 ,g′(3)=−9, and F(x)=g(x) h(x)

 Find dF/dx. x=3. 

Answer 1

I'm unsure about what you mean regarding the "h" at the end of h(x) = x² + 2x - 8h. I suspect you made an error typing it and it should not be there. I will assume h(x) = x² + 2x - 8

Since F(x) is the product of two functions, to take the derivative we use the product rule (u•v)' = u'v + uv' or, in this case F'(x) = g'(x)h(x) + g(x)h'(x) BUT, since we are taking the derivative at a specific value of "x" (x = 3) then it becomes:

F'(3) = g'(3)h(3) + g(3)h'(3)

We need h(3) which is 3² + 2(3) - 8 = 7 and we need h'(3). h'(x) = 2x + 2 (using the power rule) so h'(3) = 2(3) + 2 = 8

So F'(3) = g'(3)h(3) + g(3)h'(3) = (-9)(7) + (2)(8) = -63 + 16 = -47

If indeed, the "h" is in the function h(x), your answer would be in terms of "h"