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Find the derivative of f(x) = x^3 * ln(x). (2023)
Find the derivative of f(x) = x^3 * ln(x). (2023)
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Practice Questions
1 question
Q1
Find the derivative of f(x) = x^3 * ln(x). (2023)
3x^2 * ln(x) + x^2
3x^2 * ln(x) + x^3/x
3x^2 * ln(x) + x^3
3x^2 * ln(x) + 1
Show Solution
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Using the product rule, f'(x) = (x^3)' * ln(x) + x^3 * (ln(x))' = 3x^2 * ln(x) + x^2.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the derivative of f(x) = x^3 * ln(x). (2023)
Solution:
Using the product rule, f'(x) = (x^3)' * ln(x) + x^3 * (ln(x))' = 3x^2 * ln(x) + x^2.
Steps: 9
Show Steps
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.
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