Find the 2nd Derivative 3e^(-x^2)

Problem

d2()/(d(x)2)*3*e(−x2)

Solution

1. Identify the function and the need for the chain rule to find the first derivative.

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

(d(3)*e(−x2))/d(x)=3*e(−x2)⋅(−2*x)

3. Simplify the first derivative expression.

(d(3)*e(−x2))/d(x)=−6*x*e(−x2)

4. Apply the product rule to find the second derivative, where the two functions are u=−6*x and v=e(−x2)

(d2(3)*e(−x2))/(d(x)2)=(d(−)*6*x)/d(x)⋅e(−x2)+(−6*x)⋅d(e(−x2))/d(x)

5. Differentiate the individual components.

(d2(3)*e(−x2))/(d(x)2)=−6⋅e(−x2)+(−6*x)⋅(−2*x*e(−x2))

6. Simplify the terms by multiplying and combining like factors.

(d2(3)*e(−x2))/(d(x)2)=−6*e(−x2)+12*x2*e(−x2)

7. Factor out the common term 6*e(−x2) to reach the final form.

(d2(3)*e(−x2))/(d(x)2)=6*e(−x2)*(2*x2−1)

Final Answer

(d2(3)*e(−x2))/(d(x)2)=6*e(−x2)*(2*x2−1)

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