# (partial) Derivative of norm of vector with respect to norm of vector

I'm doing a weird derivative as part of a physics class that deals with quantum mechanics, and as part of that I got this derivative:

$$\frac{\partial}{\partial r_1} r_{12}$$

where $r_1 = |\vec r_1|$ and $r_{12} = |\vec r_1 - \vec r_2|$. Is there any way to solve this? My first guess was to set it equal to 1 or since $r_{12}$ is just a scalar, but then I realized it really depends on $r_1$ after all.

The expression appears when I try to solve

$$\frac{\nabla_1^2}{2} \left( \frac{r_{12}}{2(1+\beta r_{12})} \right)$$

($\beta$ is constant)

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Don't play dice with derivatives. – copper.hat Feb 27 '13 at 8:26
I think I'm onto a solution, by replacing r_12 with \sqrt{(x_1 - x_2)^2 + \cdots}. I'll see if I can format it nicely and post it here in a while. – Filip Sund Feb 27 '13 at 9:26
I can't answer my own question yet, so if anyone want to see the solution it's here until I can post it (in 6 hours), or someone else posts it as an answer: pastebin.com/Q4n4TYyR – Filip Sund Feb 27 '13 at 9:34
The second question, about the Laplacian of a certain function, makes sense to me. The first one does not. One can find the gradient of $r_{12}$, which is a vector of partial derivatives with respect to the components of the vector $r_1$. In any case, using the same symbol for two different things, like you did with $r_1$, is a recipe for confusion. – user53153 Feb 27 '13 at 14:01

The OP provided his solution on pastebin, which I reproduce below (it is correct).

A better way to obtain the same result would be to recall that the directional derivative is the dot product with the gradient. The gradient of the norm being $\dfrac{\vec r_1-\vec r_2}{r_{12}}$, we obtain $$\frac{\vec r_1}{r_1}\cdot \frac{\vec r_1-\vec r_2}{r_{12}}$$ which is the same answer.

\begin{align*} \frac{\partial}{\partial r_1} r_{12} &= \frac{\partial}{\partial r_1} \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2} \\ &= \frac{\partial}{\partial r_1} \sqrt{(r_1\sin\theta_1\cos\phi_1 - x_2)^2 + (r_1\sin\theta_1\sin\phi_1 - y_2)^2 + (r_1\cos\theta_1 - z_2)^2} \\ &= \left(\frac{1}{2r_{12}}\right) \frac{\partial}{\partial r_1} \Big[ (r_1\sin\theta_1\cos\phi_1 - x_2)^2 + (r_1\sin\theta_1\sin\phi_1 - y_2)^2 + (r_1\cos\theta_1 - z_2)^2 \Big] \\ &= \frac{1}{2r_{12}} \Big[ 2(x_1 - x_2)(\sin\theta_1\cos\phi_1) + 2(y_1 - y_2)(\sin\theta_1\sin\phi_1) + 2(z_1 - z_2)(\cos\theta_1) \Big] \\ &= \frac{1}{r_{12}} \left[ (x_1 - x_2)\frac{x_1}{r_1} + (y_1 - y_2)\frac{y_1}{r_1} + (z_1 - z_2)\frac{z_1}{r_1} \right] \\ &= \frac{1}{r_1 r_{12}} \Big[ r_1^2 - x_1x_2 - y_1y_2 - z_1z_2 \Big] \\ &= \frac{1}{r_1 r_{12}} \Big[ r_1^2 - \vec r_1 \vec r_2 \Big] \\ \end{align*}

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