Differentiation

Given:

Step 1: Find the first derivative using chain rule.

But , so:

Step 2: Differentiate again to get the second derivative

Use quotient rule:

Now substitute :

Step 3: Multiply both sides by to eliminate the denominator:

Step 4: Rearranging:

Or bring all terms to LHS:

But this is not matching the given options yet. Let's try expressing the final form in terms of :

Recall:

So:

Divide the whole equation by :

Final Result:

This matches option (1):

Answer:

We are given the function , and we need to find .

Step 1: Interpreting the integral The function is defined as an integral with respect to the variable , but it is evaluated with as the integrand. The integral has no explicit limits given, so it is likely a standard Gaussian-type integral, where the integrand is multiplied by a factor .

Step 2: Differentiating To differentiate , we need to use the Leibniz rule for differentiating under the integral sign. Since the integrand has no explicit dependence on except for the factor , we differentiate with respect to as follows: By the Leibniz rule, the derivative with respect to is:

Step 3: Evaluating at Now, we substitute into the expression for : We know that: Substituting this into the expression:

Conclusion Thus, the value of is: