Hello, turtle!
This problem is a killer . . .
Find the second derivative of: .
We have: .
Product Rule: .
. . . . Factor: .
Product Rule: .
. . . . . . . . . .
Factor: .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
You can also try a different approach that is sometimes helpful, but maybe not in this case which is pretty complicated.
Start with .
Take the natural log of both sides:
Expand the logs on the right side:
Take the derivative (noting the chain rule):
Move the y over:
Now take the second derivative, applying the product rule on the right side:
Substitute in the y' which was found one step earlier:
Factor out the y and simplify the garbage that's left over:
Substitute in y (which was the original equation):
Oh look, that x^2 can get passed in for a little simplification.
Also, a 2 can get pulled out:
In hindsight, this seems to be no less complicated than the first solution provided. Sometimes this technique can save considerable time though, and other times it can over complicate. I like this one though because you can see how with an exponential, the derivative is closely related to the function itself (I enjoy differential equations).