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

Problem

(d(x5)*ex)/d(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u=x5 and v=ex

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(x5)/d(x)=5*x4 using the power rule and d(ex)/d(x)=ex

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

(d(x5)*ex)/d(x)=x5d(ex)/d(x)+exd(x5)/d(x)

5. Simplify the expression by performing the multiplication.

(d(x5)*ex)/d(x)=x5*ex+5*x4*ex

6. Factor out the greatest common factor, which is x4*ex

(d(x5)*ex)/d(x)=x4*ex*(x+5)

Final Answer

(d(x5)*ex)/d(x)=x4*ex*(x+5)

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