Differentiate for D, L, σ, and T I am currently getting a primer in Calculus before taking some classes this fall and am reading Calculus Made Easy by Silvanus P. Thompson. It's a great and very readable book so far. I'm doing the exercises for chapter two on differentiation with constants and can't seem to work through this one problem. The problem is this:
The frequency $n$ of vibration of a string of diameter $D$, length $L$ and specific gravity $σ$, stretched with a force $T$, is given by
$$n=\frac1{DL}\sqrt{\frac{gT}{\pi\sigma}}.$$
Find the rate of change of the frequency when $D$, $L$, $σ$, and $T$ are varied singly. I know that I am supposed to find $dn/dD$, $dn/dL$, etc. I am, however, unsure how to approach differentiating for a single variable. Any help would be much appreciated! Thanks everyone.
 A: To take a derivative with respect to one variable just treat the others as constants. Taking the derivative of $\frac{1}{x}$ it's simpler to look at it as $x^{-1}$. Since the derivative of anything is $nx^{n-1}$ you can see that you get $\frac{1}{x^2}$. 
In your case all you have to do is break the $\frac{1}{D}$ away from the other stuff, which is just constants. 
Also, one thing I always do that makes stuff like this easier is change an equation like that to $$n = \frac{1}{D}(\frac{gT}{\sigma \pi})^{1/2}$$ or if you want to fo even further $$n = D^{-1}(\frac{gT}{\sigma \pi})^{1/2}$$
which might make it easier to see the exponents you are working with. Taking a derivative $(\frac{dn}{dD})$ of that should get you:
$$(\frac{dn}{dD}) = -D^{-2}(\frac{gT}{\sigma \pi})^{1/2}$$
One of the interesting properties of fractions and taking derivatives is that the exponents go up, not down. Another funky thing that happens is when you start doing integrals because the integral of 1/x is ln(x). But that's another story. 
EDIT: just realized i had the equation wrong myself but fixed it. 
