Finding the Derivative of h(x)=(1+f(3x))^2 using the Chain Rule

Let f be a differentiable function. If h(x)=(1+f(3x))2, which of the following gives a correct process for finding h′(x) ?

To find the derivative of the function h(x)=(1+f(3x))^2, we can use the chain rule

To find the derivative of the function h(x)=(1+f(3x))^2, we can use the chain rule.

The chain rule states that if we have a function g(x) = (f(u))^n, where f(u) is a composite differentiable function and n is a constant, then the derivative of g(x) is given by:

g'(x) = n(f(u))^(n-1) * f'(u) * u’

In this case, our function g(x) is h(x) and f(u) is 1 + f(3x). Let’s break down the steps to find h'(x) using the chain rule:

Step 1: Identify the inner function and the outer function.
The inner function is u = 3x. The outer function is f(u) = 1 + f(3x).

Step 2: Find the derivative of the inner function with respect to x.
The derivative of u = 3x with respect to x is u’ = 3.

Step 3: Find the derivative of the outer function with respect to u.
The derivative of f(u) = 1 + f(3x) with respect to u is f'(u).

Step 4: Substitute the derivatives in the chain rule formula.
Using the chain rule formula, we have:
h'(x) = 2(1 + f(3x))^(2-1) * f'(u) * u’

Step 5: Simplify the expression.
Simplifying further, we have:
h'(x) = 2(1 + f(3x)) * f'(u) * 3

Therefore, the correct process for finding h'(x) is expressed by the following option:
h'(x) = 2(1 + f(3x)) * f'(u) * 3.

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