How to Find the Derivative Using the Chain Rule | Step-by-Step Guide for Math Students

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 h′(x) of the function h(x) = (1 + f(3x))^2, we need to use the chain rule

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

The chain rule states that if we have a composition of two functions, such as h(x) = g(f(x)), then the derivative of h(x) is given by h′(x) = g′(f(x)) * f′(x).

In this case, f(3x) is the inner function within the parentheses, and 1 + f(3x) is the outer function that is squared. So, applying the chain rule to find h′(x), we need to find the derivatives of both the outer and inner functions.

Let’s break it down step by step:

1. Outer Function: (1 + f(3x))^2
The derivative of (1 + f(3x))^2 can be found using the power rule for differentiation. We multiply by the derivative of the function inside the parentheses.

The derivative of (1 + f(3x))^2 is:
2(1 + f(3x)) * (d/dx) (1 + f(3x))

2. Inner Function: f(3x)
To find the derivative of f(3x), we need to multiply by the derivative of the inner function and apply the chain rule.

The derivative of f(3x) is:
f'(3x) * (d/dx) (3x)

3. Putting it all together:
Now, we can combine the results from the outer and inner functions to find h′(x).

h′(x) = 2(1 + f(3x)) * (d/dx) (1 + f(3x))
= 2(1 + f(3x)) * f'(3x) * (d/dx) (3x)

Simplifying further, we get:
h′(x) = 2f'(3x)(1 + f(3x)) * 3

So, the correct process for finding h′(x) is to apply the chain rule as shown above.

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