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

Problem

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

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x^4 and e^x use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions to the variables u and v Let u=x^4 and v=e^x

3. Differentiate each part individually. The derivative of x^4 is 4*x^3 using the power rule, and the derivative of e^x is e^x

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

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

5. Substitute the derivatives into the expression.

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

6. Simplify the expression by factoring out the common terms x^3 and e^x

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

Final Answer

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

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