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If f(x) = ln(x^2 + 1), what is f'(x)?

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Question: If f(x) = ln(x^2 + 1), what is f\'(x)?

Options:

  1. 2x/(x^2 + 1)
  2. 1/(x^2 + 1)
  3. 2/(x^2 + 1)
  4. x/(x^2 + 1)

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

Solution: 

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

If f(x) = ln(x^2 + 1), what is f'(x)?

Practice Questions

Q1
If f(x) = ln(x^2 + 1), what is f'(x)?
  1. 2x/(x^2 + 1)
  2. 1/(x^2 + 1)
  3. 2/(x^2 + 1)
  4. x/(x^2 + 1)

Questions & Step-by-Step Solutions

If f(x) = ln(x^2 + 1), what is f'(x)?
  • Step 1: Identify the function f(x) = ln(x^2 + 1).
  • Step 2: Recognize that we need to find the derivative f'(x).
  • Step 3: Use the chain rule for differentiation. The chain rule states that if you have a function inside another function, you take the derivative of the outer function and multiply it by the derivative of the inner function.
  • Step 4: The outer function is ln(u), where u = x^2 + 1. The derivative of ln(u) is 1/u.
  • Step 5: The inner function is u = x^2 + 1. The derivative of u with respect to x is 2x.
  • Step 6: Apply the chain rule: f'(x) = (1/(x^2 + 1)) * (2x).
  • Step 7: Simplify the expression: f'(x) = 2x/(x^2 + 1).
  • Differentiation of Logarithmic Functions – The question tests the ability to differentiate a logarithmic function using the chain rule.
  • Chain Rule Application – It assesses the understanding of applying the chain rule correctly when differentiating composite functions.
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