Suppose h ( x ) = √(f ( x )) and the equation of the tangent line to 𝑓(𝑥) at 𝑥=1 is y=4+1(x-1). Find ℎ′(1)

Answer 1

h'(x) = (1/2)(f(x))^-1/2f'(x)

Tangent line for f(x) when x = 1 is y = x + 3. Slope of tangent line = f'(1) = 1. Also, the tangent line intersects the graph of f when x = 1. So, f(1) = 1 + 3 = 4.

So, h'(1) = (1/2)(f(1))^-1/2f'(1) = (1/2)(4)^-1/2(1) = 1/4