chain rule question

Question

if h(x)= sqrt(7+6*f(x)), where f(3) =7, f'(3)=4, find h'(3)

Answer 1

h(x) = sqrt(7 + 6*f(x)) can be written as:

h(x) = (7 + 6*f(x))^1/2

then we can take the derivative using the power rule and the chain rule so

h'(x) = [1/2(7 + 6*f(x))^-1/2](6)(f '(x))

So:

h'(3) = [1/2(7 + 6*f(3))^-1/2](6)(f '(3))

h'(3) = [1/2(7 + 6*7)^-1/2](6)(4)

h'(3) = [1/2(49)^-1/2](24)

h'(3) = [1/2(1/7)](24)

h'(3) = 24/14 = 12/7