Say we have some function y of x. We know
what is meant by dy by dx. This is the
derivative of the function. But this is
itself a function of x. So if we wanted
we could differentiate it again.
That is, we could differentiate the
derivative. In doing so you would obtain
what is known as the second derivative
of y. In fact, depending on the function,
we can potentially differentiate it over
and over again to obtain the third
derivative the fourth derivative and so
on. As we saw in a previous video we can
read dy by dx as d by dx of y, where d by
dx is a differential operator. I.e. it tells
us we have to differentiate y. So if we
wanted to take the derivative of the
derivative this will be d by dx of d by
dx of y.  We can simplify this by writing
it as d 2 y by dx squared. Let's try and
compute an example of a second
derivative. If we take the function y is
3x to the power of 4 plus 2 sine x, then
we know that dy by dx is 4 times 3x
cubed plus 2 times cos x, which is 12x
cubed plus 2 cos x. Now let's compute the
second derivative d 2 y by dx squared.
This is equal to d by dx of dy by dx, so
we just have to differentiate the
derivative we found. This is equal to 3
times 12x squared minus 2 times sine x,
which is 36 x squared minus 2 sine x.
