Find the Derivative - d/dx x^7e^x

Problem

d()/d(x)*x7*ex

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=x7 and v(x)=ex

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

3. Differentiate the individual components: d(x7)/d(x)=7*x6 using the power rule and d(ex)/d(x)=ex

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

(d(x7)*ex)/d(x)=x7d(ex)/d(x)+exd(x7)/d(x)

5. Simplify the expression by performing the multiplication.

(d(x7)*ex)/d(x)=x7*ex+7*x6*ex

6. Factor out the greatest common factor, which is x6*ex

(d(x7)*ex)/d(x)=x6*ex*(x+7)

Final Answer

(d(x7)*ex)/d(x)=x6*ex*(x+7)

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