Partial derivatives Partial derivatives Let w = f (x, y) be a function of two variables. Its graph is a surface in xyz-space, as w pictured. w=f(x,y) Fix a value y = y0 and just let x vary. You get a function of one variable, (1) w = f (x, y0 ), w=f(x,y 0) the partial function for y = y0 . P Its graph is a curve in the vertical plane y = y0 , whose slope at the point P where x = x0 is given by the derivative y0 (2) � � d f (x, y0 )�� , dx x0 or � ∂f �� . ∂x �(x0 ,y0 ) x0 x We call (2) the partial derivative of f with respect to x at the point (x0 , y0 ); the right side of (2) is the standard notation for it. The partial derivative is just the ordinary derivative of the partial function — it is calculated by holding one variable ﬁxed and diﬀerentiating with respect to the other variable. Other notations for this partial derivative are � � � � � ∂w �� ∂f ∂w fx (x0 , y0 ), , , ; ∂x �(x0 ,y0 ) ∂x 0 ∂x 0 the ﬁrst is convenient for including the speciﬁc point; the second is common in science and engineering, where you are just dealing with relations between variables and don’t mention the function explicitly; the third and fourth indicate the point by just using a single subscript. Analogously, ﬁxing x = x0 and letting y vary, we get the partial function w = f (x0 , y), whose graph lies in the vertical plane x = x0 , and whose slope at P is the partial derivative of f with respect to y; the notations are � � � � � � ∂f �� ∂w �� ∂f ∂w , f (x , y ), , , . y 0 0 ∂y �(x0 ,y0 ) ∂y �(x0 ,y0 ) ∂y 0 ∂y 0 The partial derivatives ∂f /∂x and ∂f /∂y depend on (x0 , y0 ) and are therefore functions of x and y. Written as ∂w/∂x, the partial derivative gives the rate of change of w with respect to x alone, at the point (x0 , y0 ): it tells how fast w is increasing as x increases, when y is held constant. For a function of three or more variables, w = f (x, y, z, . . . ), we cannot draw graphs any more, but the idea behind partial diﬀerentiation remains the same: to deﬁne the partial derivative with respect to x, for instance, hold all the other variables constant and take the ordinary derivative with respect to x; the notations are the same as above: � � � � d ∂f ∂w f (x, y0 , z0 , . . . ) = fx (x0 , y0 , z0 , . . . ), , . dx ∂x 0 ∂x 0 y