Partial derivatives

A partial derivative measures how a function changes as one input variable changes while all other input variables are held fixed. To find $\frac{\partial f}{\partial x}$, treat $y$ as a constant and differentiate with respect to $x$ using ordinary rules. To find $\frac{\partial f}{\partial y}$, treat $x$ as a constant and differentiate with respect to $y$. For example, if $f(x,y)=x^{2}y$, then $\frac{\partial f}{\partial x}=2xy$ and $\frac{\partial f}{\partial y}=x^2$. Second partial derivatives are found by differentiating a first partial derivative again; $f_{xy}$ means differentiate first with respect to $x$, then with respect to $y$.