Find the derivative of f(x) = x^3 * ln(x). (2023)

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Question: Find the derivative of f(x) = x^3 * ln(x). (2023)

Options:

Correct Answer: 3x^2 * ln(x) + x^2

Exam Year: 2023

Solution:

Using the product rule, f\'(x) = (x^3)\' * ln(x) + x^3 * (ln(x))\' = 3x^2 * ln(x) + x^2.

Find the derivative of f(x) = x^3 * ln(x). (2023)

Find the derivative of f(x) = x^3 * ln(x). (2023)

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

Product Rule – The product rule is used to find the derivative of a product of two functions, stating that (uv)' = u'v + uv'.
Chain Rule – The chain rule is used to differentiate composite functions, which may be relevant when differentiating ln(x).
Logarithmic Differentiation – Understanding how to differentiate logarithmic functions, particularly ln(x), is essential.