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) (or f_x(x,y), without the sign '). 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 f 'x(x,y) = lim Δx→0 (x+Δx)y2-xy2 Δx   = lim Δx→0 Δxy2 Δx   = y2

f 'y(x,y) = lim Δx→0 x(y+Δx)2-xy2 Δx   = lim Δx→0 2xyΔx+xΔx2 Δx   = lim Δx→0 2xy + xΔx   = 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:

Mentre nel caso univariato la derivabilità implica la continuità, ciò non accade nel caso plurivariato. Innanzitutto occorre precisare il concetto di continuità in questo caso. Se f è una funzione a più input ed 1 output:
    lim _{P → Q} f(p) = L  (se Q è interno al dominio di f o se sta sul bordo di esso)  quando per ogni ε positivo esiste un δ tale che |f(P)−L| < ε ogniqualvolta la distanza tra P e Q è minore di δ.  Se Q sta nel dominio di f e L = f(Q) si dice che f è continua in Q.
    f può ammettere in un punto Q del suo dominio tutte le derivate parziali del primo ordine senza essere ivi continua. Vedi un esempio (grafico realizzato con R).

• Let z = f(x,y).

  The partial derivative f '_x(x,y) can also be written as

  ∂f ∂x (x,y)  or  D1(f)(x,y)  or   ∂z ∂x .

  Similarly, f '_y(x,y) can also be written as

  ∂f ∂y (x,y)  or  D2(f)(x,y)  or   ∂z ∂y .

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

  f 'x(x0,y0)    or     ∂f ∂x ∣ ∣ ∣ (x0,y0)   or   D1(f)(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 / |  \ ∂f ∂x \  | / = ∂2 f ∂x2        fxy = ∂ ∂y / |  \ ∂f ∂x \  | / = ∂2 f ∂y ∂x fyy = ∂ ∂y / |  \ ∂f ∂y \  | / = ∂2 f ∂y2       fyx = ∂ ∂x / |  \ ∂f ∂y \  | / = ∂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  (controesempio)

  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.

  Notazioni.  Invece di, ad es., f_xy(x,y) e f_yy(x,y), si usano anche D_x,y(f)(x,y) e D_y,y(f)(x,y), o - più correttamente (poichè non sono obbligato a indicare con x ed y le due variabili indipendenti) - D₁₂(f)(x,y) e D₂₂(f)(x,y), o f '_{1 2}(f)(x,y) e f '_{2 2}(f)(x,y), o f_{1 2}(f)(x,y) e f_{2 2}(f)(x,y).

  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.