TITLE: derivative of function
QUESTION [5 upvotes]: I have a simple problem from calculus topics. Suppose we have
$$x=at^2,\qquad y=2at$$
and want to find $\dfrac{d^2y}{dx^2}$. There is given sample  http://www.mathopolis.com/questions/a.php?id=137&ansno=957
I think that the answer is zero  but here it is equal  D  why?

REPLY [3 votes]: Observe that $y^2 = 4ax$
Then $2y\frac{dy}{dx} = 4a$
So $\frac{dy}{dx} = \frac{2a}{y}$ [A]
Then $\frac{\mathrm d^2 y}{\mathrm dx^2} = \frac{-2a}{y^2} \frac{dy}{dx} =$ (using [A]) $-\frac{4a^2}{y^3} = -\frac{4a^2}{8a^3t^3} = -\frac{1}{2at^3}$