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 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 (x0,y0) is denoted by
(∂z/∂x) (x0, y0) , fx (x0, y0) or zx (x0, y0),
Similarly, partial derivative of z = f (x,y) with respect to y evaluated at (x0,y0) is denoted by
(∂z/∂y) (x0, y0) , fy (x0, y0) or zy (x0, y0),
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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 (x0,y0) is denoted by
(∂z/∂x) (x0, y0) , fx (x0, y0) or zx (x0, y0),
Similarly, partial derivative of z = f (x,y) with respect to y evaluated at (x0,y0) is denoted by
(∂z/∂y) (x0, y0) , fy (x0, y0) or zy (x0, y0),
For more help in Partial Derivatives click the button below to submit your homework assignment