Calculate the derivative of f(x) = x^2 * e^x.

₹0.0

Question: Calculate the derivative of f(x) = x^2 * e^x.

Options:

Correct Answer: (x^2 + 2x)e^x

Solution:

Using the product rule, f\'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.

Calculate the derivative of f(x) = x^2 * e^x.

Calculate the derivative of f(x) = x^2 * e^x.

Step 1: Identify the function f(x) = x^2 * e^x. This is a product of two functions: u = x^2 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: Calculate the derivative of u = x^2. The derivative u' = 2x.
Step 4: Calculate 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' = (2x)(e^x) + (x^2)(e^x).
Step 6: Factor out e^x from both terms: f'(x) = e^x(2x + x^2).
Step 7: Rearrange the expression: f'(x) = (x^2 + 2x)e^x.

Product Rule – The product rule is used to differentiate functions that are products of two or more functions. It states that if you have two functions u(x) and v(x), then the derivative of their product is u'v + uv'.
Exponential Functions – Understanding how to differentiate exponential functions, such as e^x, is crucial, as their derivative is the function itself.