# Product rule for the second derivative

• Nov 8th 2012, 06:15 AM
blaisem
Product rule for the second derivative
Hello. I have a function d2(xy)/dy2.

Do I separate the function as (d/dy)(d(xy)/dy) or do I just perform the product rule and apply the double derivative immediately (see below)?

Namely, the former will leave me with

(d/dy)[(y*dx/dy)+x] = dx/dy + y*d2x/dy2 + dx/dy = 2dx/dy+y*d2x/dy2

While the latter will leave me with:

d2(xy)/dy2 = y*d2x/dy2 + x*d2y/dy2 = y*d2x/dy2

I suppose I am just unsure how the product rule is to be applied in the case of a double derivative. Thank you for any advice!
• Nov 8th 2012, 09:42 AM
tom@ballooncalculus
Re: Product rule for the second derivative
The former, for sure. The 'rules' for differentiation don't involve any second derivatives.
• Nov 8th 2012, 03:20 PM
Prove It
Re: Product rule for the second derivative
Quote:

Originally Posted by blaisem
Hello. I have a function d2(xy)/dy2.

Do I separate the function as (d/dy)(d(xy)/dy) or do I just perform the product rule and apply the double derivative immediately (see below)?

Namely, the former will leave me with

(d/dy)[(y*dx/dy)+x] = dx/dy + y*d2x/dy2 + dx/dy = 2dx/dy+y*d2x/dy2

While the latter will leave me with:

d2(xy)/dy2 = y*d2x/dy2 + x*d2y/dy2 = y*d2x/dy2

I suppose I am just unsure how the product rule is to be applied in the case of a double derivative. Thank you for any advice!

First of all, are you doing full derivatives or partial derivatives?
• Nov 8th 2012, 03:45 PM
HallsofIvy
Re: Product rule for the second derivative
You have the function is xy and you are differentiating with respect to x, apparently x is a function of y. In that case, the product rule says that
$\frac{dxy}{dy}= \frac{dx}{dy} y+ x$
and then $\frac{d^2xy}{dy^2}= \frac{d}{dy}\left(\frac{dx}{dy}y+ x\right)= \frac{d^2 x}{dy^2}y+ \frac{dx}{dy}+ \frac{dx}{dy}= y\frac{d^2x}{dy^2}+ \frac{dx}{dy}$

If x is a variable independent of y, and these are partial derivatives, $\frac{\partial(xy)}{\partial y}= x$, which is independent of y so the second partial derivative, with respect to y both times, would be 0.
• Nov 11th 2012, 08:57 AM
blaisem
Re: Product rule for the second derivative
Thank you all for the advice. x was not a function of y, but I appreciate all of the replies.

I believe these were full integrals. However, it was good of you both to point out the possibility that they may have been partial derivatives. Albeit the notation in the assignment uses delta symbols, but the context is solving the commutator of two quantum mechanical operators, which are:

[x,P2y] , where the terms in brackets are operators defined as

x (hat) = x

Py(hat) = $\frac{h}{2(Pi)i} \frac{\partial}{\partial(y)}$

I figured in this context that despite the partial derivative notation, the x corresponded to a vector, so it should be treated as a full integral. I will be sure to include both possibilities in my solution. Thank you again.