# What are the differences between these two partial derivatives?

given that $z=f(x,y)$ and $x =r\cos(\theta)$ and y = $r\sin(\theta)$

$d^2(z)/d(r)^2$ and $d^2(z)/d(r)$

are both second derivatives of the function $z$? I am getting a little confused with all the notations.

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The notation for the first means $$\frac{\partial^2 z}{\partial r^2} = \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial r} \right),$$ so you take the partial derivative with respect to $r$ of the partial derivative of $z$ with respect to $r$, hence the "second derivative" of $z$ with respect to $r$.