can somebody please help me find the second derivative of y = x squared times e to the 1/x squared

thanks

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- Nov 12th 2006, 03:25 PMturtlederivatives
can somebody please help me find the second derivative of y = x squared times e to the 1/x squared

thanks - Nov 12th 2006, 05:44 PMSoroban
Hello, turtle!

This problem is a killer . . .

Quote:

Find the second derivative of: .

We have: .

Product Rule: .

. . . . Factor: .

Product Rule: .

. . . . . . . . . .

Factor: .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

- Nov 12th 2006, 06:18 PMSoltras
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). - Nov 13th 2006, 08:02 AMSoroban
I like it, Soltras!

I considered log differentiation but decided to do it head-on.

I especially like the part where becomes . . . delightful!