What is the derivative of f(x) = ln(x^2 + 1)?
Correct Answer: 2x/(x^2 + 1)
- Step 1: Identify the function we want to differentiate, which is f(x) = ln(x^2 + 1).
- Step 2: Recognize that we need to use the chain rule 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).
- Step 4: In our case, g(u) = ln(u) where u = x^2 + 1. The derivative of g(u) is g'(u) = 1/u.
- Step 5: Now, find the derivative of h(x) = x^2 + 1. The derivative 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).
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