Differentiate f(x) = ln(x^2 + 1). (2022)

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Question: Differentiate f(x) = ln(x^2 + 1). (2022)

Options:

Correct Answer: 2x/(x^2 + 1)

Exam Year: 2022

Solution:

Using the chain rule, f\'(x) = 2x/(x^2 + 1).

Differentiate f(x) = ln(x^2 + 1). (2022)

Differentiate f(x) = ln(x^2 + 1). (2022)

Step 1: Identify the function to differentiate, which is f(x) = ln(x^2 + 1).
Step 2: Recognize that we need to use the chain rule for differentiation because we have a function inside another function (ln and x^2 + 1).
Step 3: The chain rule states that if you have a function g(h(x)), the derivative is g'(h(x)) * h'(x). Here, g(u) = ln(u) and h(x) = x^2 + 1.
Step 4: Differentiate the outer function g(u) = ln(u). The derivative is g'(u) = 1/u.
Step 5: Differentiate the inner function h(x) = x^2 + 1. The derivative is h'(x) = 2x.
Step 6: Apply the chain rule: f'(x) = g'(h(x)) * h'(x) = (1/(x^2 + 1)) * (2x).
Step 7: Simplify the expression: f'(x) = 2x/(x^2 + 1).

Differentiation – The process of finding the derivative of a function.
Chain Rule – A formula for computing the derivative of the composition of two or more functions.
Natural Logarithm – The logarithm to the base e, which is often used in calculus.