Find the Derivative - d/dx y=x^4e^x

Problem

d()/d(x)*x^4*e^x

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x^4 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 each part individually: d(x^4)/d(x)=4*x^3 and d(e^x)/d(x)=e^x

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

d(y)/d(x)=x^4d(e^x)/d(x)+e^xd(x^4)/d(x)

5. Simplify the resulting expression by performing the multiplication.

d(y)/d(x)=x^4*e^x+4*x^3*e^x

6. Factor out the common terms x^3 and e^x to reach the final form.

d(y)/d(x)=x^3*e^x*(x+4)

Final Answer

d()/d(x)*x^4*e^x=x^3*e^x*(x+4)

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