## Partial Derivatives

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# Partial Derivatives

We studied the derivative as the rate of change of functions of the type y = f (x) – that is, functions of one variable only. This section is concerned with the generalization of some of the previous concepts to functions of two variables.

For example, consider a function of two independent variables written in the form z = f (x,y). We might wish to study the rate at which z changes as x changes if y is held constant. By holding y as a constant z = f (x,y) essentially becomes a function of x alone and we can calculate its derivative with respect x. This derivative is called the partial derivative of z (or f ) with respect to x and is denoted by

∂z/∂x, ∂/∂x [f (x,y)], fx (x,y) or zx (x,y).

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

∂z/∂y, ∂/∂y [ f (x,y)], fy (x,y) or zy (x,y)

and gives the rate of change of z with respect to y, with x held constant. In terms of limits, we have the following:

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

∂z/∂x = lim

h→0 h

Provided this limit exists.

The partial derivative of z with respect to y, denoted by az/ay, is the function given by

∂z/∂y = lim

k→0 k

Provided this limit exists.

From the definition, we see that to find ∂z/∂x, we treaty y as a constant and differentiate z with respect to x in the usual way. Similarly, to find ∂z/∂y we treat x as a constant and differentiate z with respect to y.

Partial derivative of z = f (x,y) with respect to x evaluated at the point (x

(∂z/∂x)

Similarly, partial derivative of z = f (x,y) with respect to y evaluated at (x

(∂z/∂y)

For more help in

For example, consider a function of two independent variables written in the form z = f (x,y). We might wish to study the rate at which z changes as x changes if y is held constant. By holding y as a constant z = f (x,y) essentially becomes a function of x alone and we can calculate its derivative with respect x. This derivative is called the partial derivative of z (or f ) with respect to x and is denoted by

∂z/∂x, ∂/∂x [f (x,y)], fx (x,y) or zx (x,y).

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

∂z/∂y, ∂/∂y [ f (x,y)], fy (x,y) or zy (x,y)

and gives the rate of change of z with respect to y, with x held constant. In terms of limits, we have the following:

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

∂z/∂x = lim

__f (x+ h, y)- f (x,y)__h→0 h

Provided this limit exists.

The partial derivative of z with respect to y, denoted by az/ay, is the function given by

∂z/∂y = lim

__f (x, y+ k) - f (x,y)__,k→0 k

Provided this limit exists.

From the definition, we see that to find ∂z/∂x, we treaty y as a constant and differentiate z with respect to x in the usual way. Similarly, to find ∂z/∂y we treat x as a constant and differentiate z with respect to y.

Partial derivative of z = f (x,y) with respect to x evaluated at the point (x

_{0},y_{0}) is denoted by(∂z/∂x)

_{(}x_{0},_{ }y_{0)}, fx (x_{0}, y_{0}) or zx (x_{0}, y_{0}),Similarly, partial derivative of z = f (x,y) with respect to y evaluated at (x

_{0},y_{0}) is denoted by(∂z/∂y)

_{(}x_{0,}_{ }y_{0)}, fy (x_{0}, y_{0}) or zy (x_{0}, y_{0}),For more help in

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