Find the Derivative - d/d@VAR F(x)=x^5e^x

Problem

d()/d(x)*x^5*e^x

Solution

1. Identify the rule needed for the derivative. Since the function is a product of two terms, x^5 and e^x use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the function. Let u=x^5 and v=e^x

3. Differentiate each part individually.

d(x^5)/d(x)=5*x^4

d(e^x)/d(x)=e^x

4. Apply the product rule formula by substituting the parts.

(d(x^5)*e^x)/d(x)=x^5d(e^x)/d(x)+e^xd(x^5)/d(x)

(d(x^5)*e^x)/d(x)=x^5*e^x+e^x*(5*x^4)

5. Simplify the expression by factoring out the common terms x^4 and e^x

(d(x^5)*e^x)/d(x)=x^4*e^x*(x+5)

Final Answer

(d(x^5)*e^x)/d(x)=x^4*e^x*(x+5)

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