Calculus problem

Question

Let f,g and h be differentiable at x. If h(x)=f(x)/g(x) +2^x,f'(3)=-8,g'(3)=5,f(3)=-4,g(3)=1,find h'(3)

Answer 1

Let f,g and h be differentiable at x. If h(x)=f(x)/g(x) +2^x,f'(3)=-8,g'(3)=5,f(3)=-4,g(3)=1,find h'(3)

For h(x) = f(x) / g(x) + 2^x, h'(x) = [f'(x)*g(x) - f(x)*g'(x)] / [g(x)]^2 + ln(2) * 2^x

by the Quotient Rule for f(x) / g(x)

and the derivative of 2^x = ln(2) * 2^x.

h'(3) = [f'(3)*g(3) - f(3)*g'(3)] / [g(3)]^2 + ln(2) * 2^3

= [-8 * 1 - (-4) * 5] / [1]^2 + ln(2) * 8

= [-8+20] + ln(2) * 8 = 12 + 8*ln(2) or 12 + ln(2^8) by property of logarithms scalar multiplication.