I'm baffled by P-set 1G-5(b). I've checked the answer and can't make sense of it.

1. I don't see how all terms but the one provided contain a 0-factor.
2. I don't understand the reasoning behind the following:

"If u =x^p, then u^p = p!"

Does the p here refer only to the pth derivative of u, or also to the pth power of x?

FWIW, I calculated the third derivative of 1G-5(a), and found the following. (The 3rd derivative is not given in the solution sheet.):

y''' = u'''v + 3u''v' + 3u'v'' +uv'''

Thank you!

Question

I'm baffled by P-set 1G-5(b). I've checked the answer and can't make sense of it.

1. I don't see how all terms but the one provided contain a 0-factor.
2. I don't understand the reasoning behind the following:

"If u =x^p, then u^p = p!"

Does the p here refer only to the pth derivative of u, or also to the pth power of x?

FWIW, I calculated the third derivative of 1G-5(a), and found the following. (The 3rd derivative is not given in the solution sheet.):

y''' = u'''v + 3u''v' + 3u'v'' +uv'''

Thank you!

anonymous · Answer

Yeah the notation is a little bit funky here.
$$u = x ^{p}$$
is just a function u where p is just an exponent.
The other one which is:
$$u ^{(p)}$$ (notice the parentheses)
Basically means that you are taking the derivative of the function u, p times. The higher derivatives are surrounded by ( ) so you can distinguish them from just normal exponents.

And from the looks of it, you third derivative looks good.