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

Problem

(d(x4)*ex)/d(x)

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x4 and ex 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=x4 and v=ex

3. Differentiate each part individually. The derivative of x4 is 4*x3 using the power rule, and the derivative of ex is ex

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

(d(x4)*ex)/d(x)=x4d(ex)/d(x)+exd(x4)/d(x)

5. Substitute the derivatives into the expression.

(d(x4)*ex)/d(x)=x4*ex+ex*(4*x3)

6. Simplify the expression by factoring out the common terms x3 and ex

(d(x4)*ex)/d(x)=x3*ex*(x+4)

Final Answer

(d(x4)*ex)/d(x)=x4*ex+4*x3*ex

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