1. second derivative of e^(1/x)

I need help finding the second derivative of e^(1/x)

So, f'(x) = e^(1/x) * (d/dx)(1/x)

f'(x) = e^(1/x) * (d/dx)(1/x)
f'(x) = -1/(x^2) * e^(1/x)

Using the product rule where u = -1/(x^2) and v = e^(1/x)
f''(x) = -1/(x^2) * [-1/(x^2) * e^(1/x)] + e^(1/x) * [2/(x^3)]
f''(x) = 1/(x^4) * e^(1/x) + 2/(x^3) * e^(1/x)

I know I am close just can’t figure out the next step.

2. Originally Posted by nearuncertainty
I need help finding the second derivative of e^(1/x)

So, f'(x) = e^(1/x) * (d/dx)(1/x)

f'(x) = e^(1/x) * (d/dx)(1/x)
f'(x) = -1/(x^2) * e^(1/x)

Using the product rule where u = -1/(x^2) and v = e^(1/x)
f''(x) = -1/(x^2) * [-1/(x^2) * e^(1/x)] + e^(1/x) * [2/(x^3)]
f''(x) = 1/(x^4) * e^(1/x) + 2/(x^3) * e^(1/x)

I know I am close just can’t figure out the next step.
You found it correctly.
there is no next step.
If you to solve the equation $f''(x)=0$.
Start by taking $e^{1/x}$ as a common factor.