Determine the derivative of f(x) = ln(x^2 + 1).

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Question: Determine the derivative of f(x) = ln(x^2 + 1).

Options:

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

Solution:

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

Determine the derivative of f(x) = ln(x^2 + 1).

Determine 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 the inside function 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).

Chain Rule – The chain rule is used to differentiate composite functions, allowing the derivative of the outer function to be multiplied by the derivative of the inner function.
Natural Logarithm Derivative – The derivative of the natural logarithm function ln(u) is 1/u, where u is a function of x.
Quotient Rule (not needed here) – While not applicable in this case, understanding when to apply the quotient rule is important for differentiating functions that are ratios.