If f(x) = ln(x^2 + 1), find f'(1). (2022)

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Question: If f(x) = ln(x^2 + 1), find f\'(1). (2022)

Options:

Correct Answer: 1

Exam Year: 2022

Solution:

f\'(x) = (2x)/(x^2 + 1). At x = 1, f\'(1) = (2*1)/(1^2 + 1) = 1.

If f(x) = ln(x^2 + 1), find f'(1). (2022)

If f(x) = ln(x^2 + 1), find f'(1). (2022)

Step 1: Identify the function f(x) = ln(x^2 + 1).
Step 2: Find the derivative f'(x) using the chain rule. The derivative of ln(u) is (1/u) * (du/dx). Here, u = x^2 + 1.
Step 3: Calculate du/dx. Since u = x^2 + 1, the derivative du/dx = 2x.
Step 4: Substitute u and du/dx into the derivative formula: f'(x) = (1/(x^2 + 1)) * (2x).
Step 5: Simplify the expression: f'(x) = (2x)/(x^2 + 1).
Step 6: Now, find f'(1) by substituting x = 1 into the derivative: f'(1) = (2*1)/(1^2 + 1).
Step 7: Calculate the values: f'(1) = 2/2 = 1.

Differentiation of Logarithmic Functions – The question tests the ability to differentiate a logarithmic function using the chain rule.
Evaluation of Derivatives – The question requires evaluating the derivative at a specific point, which tests understanding of function behavior.