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

Practice Questions

Q1
What is the derivative of f(x) = ln(x^2 + 1) at x = 1?
  1. 0
  2. 1
  3. 1/2
  4. 1/3

Questions & Step-by-Step Solutions

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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