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

Problem

d()/d(x)*x^2*e^(−2*x)

Solution

1. Identify the rule needed for the derivative, which is the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the expression to the variables u=x^2 and v=e^(−2*x)

3. Differentiate the first part u with respect to x to get d(u)/d(x)=2*x

4. Differentiate the second part v using the chain rule to get d(e^(−2*x))/d(x)=e^(−2*x)⋅(−2)=−2*e^(−2*x)

5. Substitute these components into the product rule formula: x^2*(−2*e^(−2*x))+e^(−2*x)*(2*x)

6. Simplify the expression by factoring out common terms, specifically 2*x*e^(−2*x)

Final Answer

(d(x^2)*e^(−2*x))/d(x)=2*x*e^(−2*x)*(1−x)

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