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

Problem

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

Solution

1. Identify the rule needed for the derivative. Since the function is a product of two terms, x5 and ex 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=x5 and v=ex

3. Differentiate each part individually.

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

d(ex)/d(x)=ex

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

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

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

5. Simplify the expression by factoring out the common terms x4 and 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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