Find the 2nd Derivative e^(-x^2)

Problem

d2(e(−x2))/(d(x)2)

Solution

1. Identify the function as ƒ(x)=e(−x2) and recognize that finding the second derivative requires applying the chain rule twice.

2. Apply the chain rule to find the first derivative by differentiating the exponent −x2 and multiplying it by the original exponential function.

d(e(−x2))/d(x)=e(−x2)⋅(d(−)*x2)/d(x)

d(e(−x2))/d(x)=−2*x*e(−x2)

3. Apply the product rule to the first derivative, where the two functions are u=−2*x and v=e(−x2)

d2(e(−x2))/(d(x)2)=(d(−)*2*x)/d(x)⋅e(−x2)+(−2*x)⋅d(e(−x2))/d(x)

4. Differentiate the individual components.

(d(−)*2*x)/d(x)=−2

d(e(−x2))/d(x)=−2*x*e(−x2)

5. Substitute these results back into the product rule expression.

d2(e(−x2))/(d(x)2)=−2*e(−x2)+(−2*x)*(−2*x*e(−x2))

6. Simplify the expression by performing the multiplication and factoring out the common exponential term.

d2(e(−x2))/(d(x)2)=−2*e(−x2)+4*x2*e(−x2)

d2(e(−x2))/(d(x)2)=(4*x2−2)*e(−x2)

Final Answer

d2(e(−x2))/(d(x)2)=(4*x2−2)*e(−x2)

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