Find the Derivative - d/dx y=xe^x

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the expression x*ex is a product of two functions, u=x and v=ex the product rule must be used.

2. State the product rule formula. The derivative of a product u*v is given by:

(d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual components. Find the derivative of each part:

d(x)/d(x)=1

d(ex)/d(x)=ex

4. Apply the product rule by substituting the components back into the formula:

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

5. Simplify the resulting expression by factoring out the common term ex

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

Final Answer

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

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