What is the derivative of f(x) = ln(x^2 + 1) at x = 0?
Practice Questions
1 question
Q1
What is the derivative of f(x) = ln(x^2 + 1) at x = 0?
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f'(x) = (2x)/(x^2 + 1), thus f'(0) = 0.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the derivative of f(x) = ln(x^2 + 1) at x = 0?
Solution: f'(x) = (2x)/(x^2 + 1), thus f'(0) = 0.
Steps: 8
Step 1: Identify the function we want to differentiate, which is f(x) = ln(x^2 + 1).
Step 2: Use the chain rule to find the derivative of f(x). 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 3: The outer function is ln(u) where u = x^2 + 1. The derivative of ln(u) is 1/u.
Step 4: The inner function u = x^2 + 1. The derivative of u with respect to x is 2x.
Step 5: Now apply the chain rule: f'(x) = (1/(x^2 + 1)) * (2x). This simplifies to f'(x) = (2x)/(x^2 + 1).
Step 6: Now we need to find the derivative at x = 0. Substitute x = 0 into f'(x).
