Find the Derivative - d/dx f(x)=xe^x

Problem

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

Solution

1. Identify the rule needed for the derivative. Since the expression is a product of two functions, x and ex the product rule must be used.

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

3. Assign the variables such that u=x and v=ex

4. Differentiate each part individually.

d(x)/d(x)=1

d(ex)/d(x)=ex

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

(d(x)*ex)/d(x)=x⋅ex+ex⋅1

6. Factor out the common term ex to simplify the expression.

(d(x)*ex)/d(x)=ex*(x+1)

Final Answer

(d(x)*ex)/d(x)=ex*(x+1)

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