Find the Derivative - d/dx f(x)=x^7e^(4x)

Problem

d()/d(x)*x7*e(4*x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=x7 and v(x)=e(4*x)

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

3. Differentiate the first part u=x7 using the power rule to get d(x7)/d(x)=7*x6

4. Differentiate the second part v=e(4*x) using the chain rule to get d(e(4*x))/d(x)=4*e(4*x)

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

(d(x7)*e(4*x))/d(x)=x7*(4*e(4*x))+e(4*x)*(7*x6)

6. Simplify the expression by factoring out the greatest common factor, x6*e(4*x)

(d(x7)*e(4*x))/d(x)=x6*e(4*x)*(4*x+7)

Final Answer

(d(x7)*e(4*x))/d(x)=x6*e(4*x)*(4*x+7)

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