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 and the product rule.

2. Find the first derivative by applying the chain rule, where the derivative of eu is eu⋅d(u)/d(x)

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 2*x*e(x2) to find the second derivative, using the formula d()/d(x)*[u*v]=ud(v)/d(x)+vd(u)/d(x)

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

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

4. Substitute the previously found derivative of e(x2) and the derivative of 2*x into the expression.

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

5. Simplify the expression by multiplying the terms and factoring out the common term 2*e(x2)

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

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

Final Answer

d2(e(x2))/(d(x)2)=2*e(x2)*(2*x2+1)

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