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

Problem

d()/d(x)*x*e(−x)

Solution

1. Identify the rule needed for differentiation. Since the function is a product of x and e(−x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the function to u and v Let u=x and v=e(−x)

3. Differentiate each part separately. The derivative of u is d(x)/d(x)=1 The derivative of v requires the chain rule: d(e(−x))/d(x)=e(−x)⋅(−1)=−e(−x)

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

(d(x)*e(−x))/d(x)=x*(−e(−x))+e(1)(−x)

5. Simplify the expression by factoring out the common term e(−x)

(d(x)*e(−x))/d(x)=−x*e(−x)+e(−x)

(d(x)*e(−x))/d(x)=e(−x)*(1−x)

Final Answer

(d(x)*e(−x))/d(x)=e(−x)*(1−x)

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