What is the derivative of f(x) = ln(x^2 + 1) at x = 1?

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Question: What is the derivative of f(x) = ln(x^2 + 1) at x = 1?

Options:

Correct Answer: 1

Solution:

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

What is the derivative of f(x) = ln(x^2 + 1) at x = 1?

What is the derivative of f(x) = ln(x^2 + 1) at x = 1?

Step 1: Identify the function we need to differentiate, which is f(x) = ln(x^2 + 1).
Step 2: Use the chain rule to find the derivative of f(x). The derivative of ln(u) is (1/u) * (du/dx), where 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, we need to find the derivative at x = 1. Substitute x = 1 into f'(x).
Step 7: Calculate f'(1) = (2*1)/(1^2 + 1) = 2/2 = 1.

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