Second derivative I have a following equation
$$\frac{dh}{dt}=\frac{2}{\pi(3\cos 2h +4)^2}$$
In my assignment I was ordered to calculate:
$\frac{d^2h}{dt^2}$
I thought that it will be enough just to calculate second derivative of $\frac{dh}{dt}$, but according to the anwer key it should be equal to $\frac{d}{dt}\Big(\frac{dh}{dt}\Big)=\frac{dh}{dt}\times \frac{d}{dh}\Big(\frac{dh}{dt}\Big)$
I am not really sure why it should be like this and would like to know general rule that is behid this example. Thank you for your help!
 A: Well the second derivative $\frac{d^2h}{dt^2}$ is done by taking the derivative of $\frac{dh}{dt}$ with respect to $t$. This is simply $\frac{d}{dt} \left(\frac{dh}{dt}\right)$. But in this particular case, note that your $\frac{dh}{dt}$ is given as a function of $h$. So how would you do $\frac{d}{dt}\left(\frac{2}{\pi(3\cos 2h + 4)^2}\right)$? The variable inside is $h$ and you're trying to differentiate with respect to $t$, this is problematic. You can't do it (not easily, at least). 
So we use the chain rule: $\displaystyle \frac{da}{dt} = \frac{da}{dh} \times \frac{dh}{dt}$ (it looks just like cancelling fractions). 
Here we have $a = \frac{dh}{dt}$. So we can differentiate $\frac{dh}{dt}$ with respect to $h$, and then multiplty that result by $\frac{dh}{dt}$. We know what $\frac{dh}{dt}$ is (it's given to us) and we know how to differentiate $\frac{dh}{dt}$ with respect to $h$ (since it's given in terms of $h$). 
Hence the easiest option here is $\displaystyle \frac{d^2h}{dt^2} = \frac{d}{dt}\left(\frac{dh}{dt}\right) = \frac{d}{dh}\left(\frac{dh}{dt}\right) \cdot \frac{dh}{dt}$. 
I presume you know how to do $\frac{2}{\pi}\frac{d}{dh}\left((3+\cos 2h+4)^{-1}\right)$.
A: Second derivative of h with respect to t is 
d /dt (dh/dt)
= d( 2/(pi(3cos2h +4)^2)/dt
(Here pi is the ratio of circumference to diameter of a circle.)
= 2/(pi ( 3 cos2h +4))× (-2)×      d( 3cos2h +4)/dt
( By chain rule)
= -4/(pi ( 3 cos2h +4))×(-3sin2h)×2dh/dt
= 24sin2h/(pi ( 3 cos2h +4))×2/(pi ( 3 cos2h +4)^2)
=48sin2h/(pi^2( 3 cos2h +4)^3).
