What is the second derivative of e^(-1/x)?

To find the second derivative of the function e^(-1/x), we need to go through a few steps. Let's get started!

Step 1: Find the first derivative
To find the first derivative, we need to apply the chain rule. Let u = -1/x, then the function becomes e^u. The derivative of e^u with respect to u is simply e^u. Now, we need to apply the chain rule. The derivative of u with respect to x is: du/dx = d(-1/x)/dx. Applying the chain rule, this becomes du/dx = (1/x^2). So, the first derivative is de^u / dx = e^u * du/dx = e^(-1/x) * (1/x^2).

Step 2: Find the second derivative
To find the second derivative, we need to take the derivative of the first derivative we found in step 1. Applying the product rule, we get:
d²e^u / dx² = (e^(-1/x) * (1/x^2))' = (e^(-1/x))' * (1/x^2) + e^(-1/x) * (1/x^2)'.

Now, let's find the derivatives of each term separately:
- The derivative of e^(-1/x) with respect to x is obtained using the chain rule: (e^(-1/x))' = (-1/x^2) * e^(-1/x).
- The derivative of (1/x^2) with respect to x is (-2/x^3).

So, the second derivative is:
d²e^u / dx² = (-1/x^2) * e^(-1/x) * (1/x^2) + e^(-1/x) * (-2/x^3).

Simplifying this expression, we have:
d²e^u / dx² = -e^(-1/x) / x^4 - 2e^(-1/x) / x^4.

Therefore, the second derivative of e^(-1/x) is given by -e^(-1/x) / x^4 - 2e^(-1/x) / x^4.