A Partial derivative in a function of several variables is the rate of change of the function, by allowing one variable to vary while freezing all the other variables to constants.
Let f(x, y) be a function of two variable x and y. Suppose y is fixed as a constant y = b and only the variable x is allowed to change. Then we are really considering a function of one variable x and we can denote this as g(x) = f(x, b). Hence the partial derivative of f(x, y) with respect to x is indeed g'(x) and denoted by f
_{x}.
Similarly the partial derivative of f(x, y) with respect to y is denoted by f
_{y} and it is equal to h'(y) where h(y) = f(y, b).
The commonly used Leibniz notation for partial derivatives are:
$\frac{\partial f}{\partial x}$ representing the partial derivative of f with respect to x and
$\frac{\partial f}{\partial y}$ representing the partial derivative of f with respect to y.Limit Definition of Partial derivativesThe partial derivative of f(x, y) with respect to x at the point (a, b) is
$\frac{\partial f}{\partial x}_{(a,b)}$ =
$\lim_{h>0}\frac{f(a+h,b)f(a,b)}{h}$The Partial derivative of f(x, y) with respect to y at the point (a, b) is
$\frac{\partial f}{\partial y}_{(a,b)}$ =
$\lim_{h>0}\frac{f(a,b+h)f(a,b)}{h}$Geometrically a partial derivative of a function of two variables can be interpreted as the slopes of the tangents to the traces of the surface represented by the function.
If S is the surface represented by the function z = f(x, y), then C_{1} and C_{2} are the traces cut off by the planes y = b and x = a at the point P(a, b, c). The partial derivatives f_{x} and fy represent slopes of the tangent lines (T_{1} and T_{2}) to the curves C_{1} and C_{2}.

