Find the Derivative - d/dx f(x)=x^2e^x

Problem

(d(x^2)*e^x)/d(x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^2 and e^x use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions for the product rule. Let u=x^2 and v=e^x

3. Differentiate each part individually. The derivative of x^2 is 2*x and the derivative of e^x is e^x

4. Apply the product rule formula by substituting the functions and their derivatives.

(d(x^2)*e^x)/d(x)=x^2d(e^x)/d(x)+e^xd(x^2)/d(x)

5. Substitute the derivatives into the expression.

(d(x^2)*e^x)/d(x)=x^2*e^x+e^x*(2*x)

6. Simplify the expression by factoring out the common term x*e^x

(d(x^2)*e^x)/d(x)=x*e^x*(x+2)

Final Answer

(d(x^2)*e^x)/d(x)=x^2*e^x+2*x*e^x

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