If
h(x) = (4 + 3f(x))^(1/2) ,where f(3) = 4 and f '(3) = 3,find h'(3).
9/8
To find h'(3), the derivative of h(x) at x = 3, we can use the chain rule because h(x) is composed of the function f(x). Here are the steps to find the derivative:
Step 1: Apply the Chain Rule: The chain rule states that if we have a composite function g(f(x)), then the derivative of g(f(x)) with respect to x is given by g'(f(x)) * f'(x).
Step 2: Find f'(x): We are given that f'(3) = 3, which means the derivative of f(x) at x = 3 is 3.
Step 3: Differentiate h(x): We need to differentiate h(x) with respect to x. Notice that h(x) is of the form g(f(x)), where g(u) = (4 + 3u)^(1/2). Applying the chain rule, we have g'(u) = (1/2)(4 + 3u)^(-1/2) * 3.
Step 4: Calculate h'(3): Plug in f'(x) and the value of f(3) into the expression for g'(u). Since we are interested in finding h'(3), we need to evaluate the derivative at x = 3.
h'(3) = g'(f(3)) * f'(3)
= (1/2)(4 + 3f(3))^(-1/2) * 3 * f'(3)
= (1/2)(4 + 3(4))^(-1/2) * 3 * 3
= (1/2)(4 + 12)^(-1/2) * 9
= (1/2)(16)^(-1/2) * 9
= (1/2)(1/4) * 9
= 1/8 * 9
= 9/8
= 1.125
Therefore, h'(3) = 1.125.