Partial Derivatives

Consider a function of two independent variables,

 Z = f(x,y)

Here, we might wish to study the rate at which z changes as x changes, if y is held constant. By keeping y as constant, z = f(x,y) naturally becomes a function of x alone and so, the derivative of the function can be calculated with respect to x. This derivative is Partial derivative of z (or f) with respect to x and is denoted by

∂z/∂x, ∂/∂x [f(x,y)],     F_x (x,y) or Z_x (x, y)

Similarly, the partial derivative of z (or f) with respect to y is denoted by

∂z/∂y, ∂/∂y [f(x,y)],  F_y (x,y) or Z_y (x, y);

Which gives the rate of change of z with respect to y keeping x as constant.

We can define partial derivatives with the help of limits as follows :

Definition

If z = f(x,y), the partial derivative of z with respect to x denoted by ∂z/∂x is the function given by

∂z/∂x = lim ƒ(x,y+k-ƒ(x,y)/k

h → 0

Provided, the above limit exists.

Similarly, the partial derivatives of z with respect to y denoted by ∂z/∂y is the function given by

∂z/∂x = lim ƒ(x,y+k)-ƒ(x,y)/k

h → 0

provided, the above limit exists.

Example : If z = 2x² + 3xy – 5y², the find

∂z/∂x and ∂z/∂y

i. In order to find  ∂z/∂x, we have to held y as a constant ad differentiate z with respect to x. Thus,

∂z/∂x =∂/∂x  (2x² + 3xy – 5y²) = 4x + 3y

(ii) Similarly, in order to find ∂z/∂y, we have to hold x as a constant, and differentiate z with respect to y. Thus,  ∂z/∂y [2x² + 3xy – 5y²] = 3x – 10y