Suppose you want to forecast the weather this weekend in Los Angeles. You construct a formula for the temperature as a function of several environmental variables, each of which is not entirely predictable. Now you would like to see how your weather forecast would change as one particular environmental factor changes, holding all the other factors constant. To do this investigation, you would use the concept of a partial derivative...

Let the temperature T depend on variables x and y, T = f(x,y). The rate of change of f with respect to x (holding y constant) is called the partial derivative of f with respect to x and is denoted by f_x(x,y). Similarly, the rate of change of f with respect to y is called the partial derivative of f with respect to y and is denoted by f_y(x,y).

Do you see the similarity beween these and the limit definition of a function of one variable?

Example

Let f(x,y) = xy2   Then fx(x,y) = lim Dx®0 (x+Dx)y2-xy2 Dx   = lim Dx®0 Dxy2 Dx   = y2

fy(x,y) = lim Dx®0 x(y+Dx)2-xy2 Dx   = lim Dx®0 2xyDx+xDx2 Dx   = lim Dx®0 2xy + xDx   = 2xy

In practice, we use our knowledge of single-variable calculus to compute partial derivatives. To calculate f_x(x,y), we view y as a constant and differentiate f(x,y) with respect to x:

• Let z = f(x,y).

  The partial derivative f_x(x,y) can also be written as

  ¶f ¶x (x,y)  or   ¶z ¶x .

  Similarly, f_y(x,y) can also be written as

  ¶f ¶y (x,y)  or   ¶z ¶y .

• The partial derivative f_y(x,y) evaluated at the point (x₀,y₀) can be expressed in several ways:

  fx(x0,y0)     ,      ¶f ¶x ê ê ê (x0,y0)      , or     ¶f ¶x (x0,y0)

  There are analogous expressions for f_y(x₀,y₀).

Suppose the graph of z = f(x,y) is the surface shown. Consider the partial derivative of f with respect to x at a point (x₀,y₀).

Holding y constant and varying x, we trace out a curve that is the intersection of the surface with the vertical plane y = y₀.

The partial derivative f_x(x₀,y₀) measures the change in z per unit increase in x along this curve. That is, f_x(x₀,y₀) is just the slope of the curve at (x₀,y₀). The geometrical interpretation of f_y(x₀,y₀) is analogous.

• Functions of More than Two Variables

  For g(x,y,z), the partial derivative g_x(x,y,z) is calculated by holding y and z constant and differentiating with respect to x. The partial derivatives g_y(x,y,z) and g_z(x,y,z) are calculated in an analagous manner.

  Example

• Higher-Order Partial Derivatives

  For a function f(x,y), the partial derivatives ¶f / ¶x and ¶f / ¶y are themselves functions of x and y, so we can take partial derivatives of them:

  fxx = ¶ ¶x æ ç è ¶f ¶x ö ÷ ø = ¶2 f ¶x2        fxy = ¶ ¶y æ ç è ¶f ¶x ö ÷ ø = ¶2 f ¶y ¶x fyy = ¶ ¶y æ ç è ¶f ¶y ö ÷ ø = ¶2 f ¶y2        fyx = ¶ ¶x æ ç è ¶f ¶y ö ÷ ø = ¶2 f ¶x ¶y

  fxy and fyx are called mixed second-order partial derivatives. If f, fx, fy, fxy, and fyx are continuous on an open region, then fxy = fyx at each point in the region, so the order in which the differentiation is done does not matter.

  Higher-order partial derivatives (e.g. f_xxy) can also be calculated. Using the subscript notation, the order of differentiation is from left to right.

  Example

Consider a function f(x,y).

To calculate f_x(x,y), differentiate f with respect to x holding y constant. Similarly, to calculate f_y(x,y), differentiate f with respect to y holding x constant.