What is the derivative of f(x) = x^3 * ln(x)? (2023)

₹0.0

Question: What is 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) = 3x^2 * ln(x) + x^2.

What is the derivative of f(x) = x^3 * ln(x)? (2023)

What is the derivative of f(x) = x^3 * ln(x)? (2023)

Step 1: Identify the function f(x) = x^3 * ln(x). This is a product of two functions: u = x^3 and v = ln(x).
Step 2: Recall the product rule for derivatives, which states that if you have two functions u and v, then the derivative f'(x) = u'v + uv'.
Step 3: Calculate the derivative of u = x^3. The derivative u' = 3x^2.
Step 4: Calculate the derivative of v = ln(x). The derivative v' = 1/x.
Step 5: Apply the product rule: f'(x) = u'v + uv' = (3x^2)(ln(x)) + (x^3)(1/x).
Step 6: Simplify the second term: (x^3)(1/x) = x^2.
Step 7: Combine the terms: f'(x) = 3x^2 * ln(x) + x^2.

No concepts available.