Using the product rule, f'(x) = 3x^2 * e^x + x^3 * e^x.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the derivative of f(x) = x^3 * e^x?
Solution: Using the product rule, f'(x) = 3x^2 * e^x + x^3 * e^x.
Steps: 7
Step 1: Identify the function f(x) = x^3 * e^x. This is a product of two functions: u = x^3 and v = e^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 + uv'.
Step 3: Find the derivative of u = x^3. The derivative u' = 3x^2.
Step 4: Find the derivative of v = e^x. The derivative v' = e^x (since the derivative of e^x is e^x).
Step 5: Apply the product rule: f'(x) = u'v + uv'. Substitute u', v, u, and v' into the formula.
Step 6: Substitute the values: f'(x) = (3x^2)(e^x) + (x^3)(e^x).
