If f(x) = x^2 * ln(x), what is f'(x)? (2022)

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Question: If f(x) = x^2 * ln(x), what is f\'(x)? (2022)

Options:

Correct Answer: 2x * ln(x) + x

Exam Year: 2022

Solution:

Using the product rule, f\'(x) = 2x * ln(x) + x.

If f(x) = x^2 * ln(x), what is f'(x)? (2022)

If f(x) = x^2 * ln(x), what is f'(x)? (2022)

Step 1: Identify the function f(x) = x^2 * ln(x).
Step 2: Recognize that this is a product of two functions: u = x^2 and v = ln(x).
Step 3: Recall the product rule for differentiation: (u * v)' = u' * v + u * v'.
Step 4: Differentiate u = x^2. The derivative u' = 2x.
Step 5: Differentiate v = ln(x). The derivative v' = 1/x.
Step 6: Apply the product rule: f'(x) = u' * v + u * v'.
Step 7: Substitute u, u', v, and v' into the product rule: f'(x) = (2x) * (ln(x)) + (x^2) * (1/x).
Step 8: Simplify the second term: (x^2) * (1/x) = x.
Step 9: Combine the terms: f'(x) = 2x * ln(x) + x.

Product Rule – The product rule is used to differentiate functions that are products of two or more functions.
Natural Logarithm Derivative – The derivative of ln(x) is 1/x, which is important when differentiating functions involving logarithms.