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

Practice Questions

Q1
What is the derivative of f(x) = x^3 * ln(x)? (2023)
  1. 3x^2 * ln(x) + x^2
  2. 3x^2 * ln(x) + x^3/x
  3. 3x^2 * ln(x) + 3x^2
  4. 3x^2 * ln(x) + 1

Questions & Step-by-Step Solutions

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.
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