partial derivative

The rate of change of a multivariable function in one variable's direction, with every other variable held fixed.

The partial derivative of $f$ with respect to $x$ at $(a,b)$ is $f_x(a,b) = \frac{\partial f}{\partial x}(a,b) = \lim_{h\to 0} \frac{f(a+h,b) - f(a,b)}{h}$ , and $f_y(a,b)$ is defined the same way, incrementing $y$ instead. To compute either one, hold every OTHER variable fixed and differentiate as an ordinary single-variable function — $\partial$ signals that other variables are present but frozen, not that anything new about differentiation itself is happening. $f_x$ and $f_y$ are themselves new functions of $(x,y)$ .

Examples

For $f(x,y)=x^2y-y^3$ , $f_x=2xy$ and $f_y=x^2-3y^2$ , so $f_x(1,1)=2$ and $f_y(1,1)=-2$ .