Find the 2nd Derivative f(x)=xe^(-x^2)

Problem

ƒ(x)=x*e(−x2)

Solution

1. Identify the function as a product of x and e(−x2) and apply the product rule, which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first time to find ƒ(x)′ using the chain rule for the exponential term.

ƒ(x)′=x⋅d(e(−x2))/d(x)+e(−x2)⋅d(x)/d(x)

ƒ(x)′=x*(e(−x2)⋅(−2*x))+e(1)(−x2)

ƒ(x)′=−2*x2*e(−x2)+e(−x2)

3. Simplify the first derivative by factoring out the common exponential term.

ƒ(x)′=e(−x2)*(1−2*x2)

4. Differentiate again to find ƒ(x)″ by applying the product rule to the simplified first derivative.

ƒ(x)″=e(−x2)⋅d(1−2*x2)/d(x)+(1−2*x2)⋅d(e(−x2))/d(x)

5. Calculate the derivatives of the individual components.

ƒ(x)″=e(−x2)*(−4*x)+(1−2*x2)*(e(−x2)⋅(−2*x))

6. Distribute and combine terms to simplify the expression.

ƒ(x)″=−4*x*e(−x2)−2*x*e(−x2)+4*x3*e(−x2)

ƒ(x)″=4*x3*e(−x2)−6*x*e(−x2)

7. Factor the final expression to reach the most concise form.

ƒ(x)″=2*x*e(−x2)*(2*x2−3)

Final Answer

ƒ(x)″=2*x*e(−x2)*(2*x2−3)

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