Product rule for the second derivative

Hello. I have a function d^{2}(xy)/dy^{2}.

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*d^{2}x/dy^{2} + dx/dy = 2dx/dy+y*d^{2}x/dy^{2 }

While the latter will leave me with:

d^{2}(xy)/dy^{2} = y*d^{2}x/dy^{2} + x*d^{2}y/dy^{2} = y*d^{2}x/dy^{2}

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!

Re: Product rule for the second derivative

The former, for sure. The 'rules' for differentiation don't involve any second derivatives.

Re: Product rule for the second derivative

Quote:

Originally Posted by

**blaisem** Hello. I have a function d^{2}(xy)/dy^{2}.

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*d^{2}x/dy^{2} + dx/dy = 2dx/dy+y*d^{2}x/dy^{2 }

While the latter will leave me with:

d^{2}(xy)/dy^{2} = y*d^{2}x/dy^{2} + x*d^{2}y/dy^{2} = y*d^{2}x/dy^{2}

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?

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

$\displaystyle \frac{dxy}{dy}= \frac{dx}{dy} y+ x$

and then $\displaystyle \frac{d^2xy}{dy^2}= \frac{d}{dy}\left(\frac{dx}{dy}y+ x\right)= \frac{d^2 x}{dy^2}y+ \frac{dx}{dy}+ \frac{dx}{dy}= y\frac{d^2x}{dy^2}+ \frac{dx}{dy}$

If x is a variable independent of y, and these are partial derivatives, $\displaystyle \frac{\partial(xy)}{\partial y}= x$, which is independent of y so the second partial derivative, with respect to y both times, would be 0.

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,P^{2}_{y}] , where the terms in brackets are operators defined as

x (hat) = x

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

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.