What is the error in the value of the Lennard-Jones potential given: $\epsilon , \sigma , r$ The Lenard-Jones Potential, models the interaction between a pair of neutral atoms or molecules, and is given by: 
$$V_{LJ}=4\epsilon \Big[\Big(\dfrac{\sigma}{r}\Big)^{12} - \Big(\dfrac{\sigma}{r}\Big)^6\Big]$$
where $\epsilon$ is the depth of the potential well, $\sigma$ is the finite distance at which the inter-particle potential is zero, and $r$ is the finite distance between particles. If the error in measured values of $\epsilon $, $\sigma$ and $r$ is $3\%$, $5\%$ and $5\%$ respectively, what is the error in the value of $V_{LJ}$.

I'm not sure of the form I should put this question in, any help is appreciated. 

 A: The variance of a function is equal to the variance of the input times the square of the derivative. If the function has multiple inputs whose errors are independent, then the variances from each input add together. So we have
\begin{multline}
\sigma^2_V = \sigma^2_\epsilon\left(\frac{\partial V}{\partial \epsilon}\right)^2 +  \sigma^2_r\left(\frac{\partial V}{\partial r}\right)^2 +  \sigma^2_\sigma\left(\frac{\partial V}{\partial \sigma}\right)^2
\\ =  \sigma^2_\epsilon\left(\frac{V}{\epsilon}\right)^2 +  \sigma^2_r\left(-\frac{24\epsilon}{r}\left[2\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]\right)^2 +  \sigma^2_\sigma\left(\frac{24\epsilon}{\sigma}\left[2\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]\right)^2
\\ = 16\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]^2\sigma_\epsilon^2 + 576\epsilon^2\left[2\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]^2\left[\left(\frac{\sigma_r}{r}\right)^2 + \left(\frac{\sigma_\sigma}{\sigma}\right)^2\right]
\end{multline}
It often simplifies things to write in terms of the percent error. Doing this with V gives
$$
\left(\frac{\sigma_V}{V}\right)^2 = \left(\frac{\sigma_\epsilon}{\epsilon}\right)^2 + 36\left[ 1+ \frac{1}{1-(r/\sigma)^6}\right]^2\left[\left(\frac{\sigma_r}{r}\right)^2 + \left(\frac{\sigma_\sigma}{\sigma}\right)^2\right]
$$
though obviously you need to watch out when using this form near $r = \sigma$.
