Find the Derivative - d/dx x^2e^(-x)

Problem

d()/d(x)*x2*e(−x)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u=x2 and v=e(−x)

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

3. Differentiate the first part u=x2 using the power rule to get d(x2)/d(x)=2*x

4. Differentiate the second part v=e(−x) using the chain rule to get d(e(−x))/d(x)=−e(−x)

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

(d(x2)*e(−x))/d(x)=x2*(−e(−x))+e(−x)*(2*x)

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

(d(x2)*e(−x))/d(x)=−x2*e(−x)+2*x*e(−x)

(d(x2)*e(−x))/d(x)=x*e(−x)*(2−x)

Final Answer

d()/d(x)*x2*e(−x)=2*x*e(−x)−x2*e(−x)

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