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 is a product of two functions, −x and e^x the product rule must be applied.

2. Apply the product rule, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) Let u=−x and v=e^x

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

4. Substitute these derivatives into the product rule formula.

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

5. Simplify the resulting expression by performing the differentiation and combining terms.

(d(−)*x*e^x)/d(x)=−x*e^x+e^x*(−1)

6. Factor out the common term e^x to reach the final simplified form.

(d(−)*x*e^x)/d(x)=−x*e^x−e^x

Final Answer

(d(−)*x*e^x)/d(x)=−e^x*(x+1)

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