What is the derivative of f(x) = x^2 * sin(x)? (2023)
Practice Questions
Q1
What is the derivative of f(x) = x^2 * sin(x)? (2023)
2x * sin(x) + x^2 * cos(x)
2x * cos(x) + x^2 * sin(x)
2x * sin(x) - x^2 * cos(x)
x^2 * sin(x) + 2x * cos(x)
Questions & Step-by-Step Solutions
What is the derivative of f(x) = x^2 * sin(x)? (2023)
Step 1: Identify the function f(x) = x^2 * sin(x). This is a product of two functions: u = x^2 and v = sin(x).
Step 2: Recall the product rule for derivatives. The product rule states that if you have two functions u and v, then the derivative f'(x) = u' * v + u * v'.
Step 3: Find the derivative of u = x^2. The derivative u' = 2x.
Step 4: Find the derivative of v = sin(x). The derivative v' = cos(x).
Step 5: Apply the product rule: f'(x) = u' * v + u * v' = (2x) * (sin(x)) + (x^2) * (cos(x)).
Step 6: Combine the results to get the final answer: f'(x) = 2x * sin(x) + x^2 * cos(x).
