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

Problem

d()/d(x)*x^2*e^(−x)

Solution

1. Identify the rule needed for the expression, which is the product of two functions: u=x^2 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=x^2 using the power rule to get d(x^2)/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(x^2)*e^(−x))/d(x)=x^2*(−e^(−x))+e^(−x)*(2*x)

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

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

(d(x^2)*e^(−x))/d(x)=x*e^(−x)*(2−x)

Final Answer

d()/d(x)*x^2*e^(−x)=2*x*e^(−x)−x^2*e^(−x)

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