# 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!

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 $\frac{2}{\pi}\frac{d}{dh}\left((3+\cos 2h+4)^{-1}\right)$.

• Excelent answer, now I understand everything. Thanks a lot! – user3641083 Apr 21 '17 at 20:42

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).