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

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

Options:

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

Solution:

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

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

Calculate the derivative of f(x) = ln(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 h(x) = x^2 + 1. The derivative of h(x) is 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, which in this case involves the natural logarithm and a polynomial.
Logarithmic Differentiation – Understanding how to differentiate logarithmic functions, particularly the natural logarithm, is essential.
Quotient Rule – While not directly used here, recognizing when to apply the quotient rule can be a related concept when dealing with derivatives.