Find the Derivative - d/dx xe^x

Problem

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

Solution

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

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

3. Differentiate the individual components. The derivative of x is 1 and the derivative of e^x is e^x

4. Substitute these values into the product rule formula.

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

5. Simplify the resulting expression by performing the differentiation and factoring if possible.

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

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

Final Answer

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

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