Find the Derivative - d/dx x^5e^x

Problem

(d(x^5)*e^x)/d(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u=x^5 and v=e^x

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

3. Differentiate the individual components: d(x^5)/d(x)=5*x^4 using the power rule and d(e^x)/d(x)=e^x

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

(d(x^5)*e^x)/d(x)=x^5d(e^x)/d(x)+e^xd(x^5)/d(x)

5. Simplify the expression by performing the multiplication.

(d(x^5)*e^x)/d(x)=x^5*e^x+5*x^4*e^x

6. Factor out the greatest common factor, which is x^4*e^x

(d(x^5)*e^x)/d(x)=x^4*e^x*(x+5)

Final Answer

(d(x^5)*e^x)/d(x)=x^4*e^x*(x+5)

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