Find the Derivative - d/dx y=x^3e^x

Problem

d()/d(x)*x^3*e^x

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x^3 and v=e^x

2. Apply the product rule, which states that d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual components: d(x^3)/d(x)=3*x^2 and d(e^x)/d(x)=e^x

4. Substitute these derivatives back into the product rule formula.

x^3⋅e^x+e^x⋅3*x^2

5. Factor out the common terms x^2 and e^x to simplify the expression.

x^2*e^x*(x+3)

Final Answer

d()/d(x)*x^3*e^x=x^2*e^x*(x+3)

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