微積分:偏微分

6.2

Partial Derivatives

Definition 30 If  =  ( )  then the first partial derivative of  with respect to  and  are the function  and  defined by

( ) = lim ∆→0  ( + ∆ )_{−  ( )} ∆ ( ) = lim ∆→0  (  + ∆)_{−  ( )} ∆

provided the limits exist.

Definition 31 The partial derivative of  ( ) with repect to  is the func-tion obtained by computing the derivative of  ( ) while treating  as a constant and  ( ) as a function of  alone. Such that a derivative can be represented by

( ) 

Similarly, the partial derivative of  ( ) with repect to  is the function obtained by computing the derivative of  ( ) while treating  as a constant and  ( ) as a function of  alone. Such that a derivative can be represented by

( ) 

Example 151 Find the partial derivatives  and  for the function

 ( ) = 2( + 7)5

Example 152 Find the partial derivatives  and  for the function

 ( ) = 32

Notation: For  =  ( )  the partial derivative  and  are defined

by   ( ) = ( ) =  =   and   ( ) = ( ) =  =  

The first partial derivative evaluated at the point ( ) are denoted by   ¯ ¯_() = ( ) and   ¯ ¯_() = ( )  82

Example 153 For the function

 = 

5_2

23_{+ 8}

find formulas for _ and _

Example 154 For  ( ) = 2find  and and evaluate each at the

point (1 ln 2) 

Remark 5 Informlly, we say that the values of   and   at the point (0 0 0) denote the slopes of the surface in the − and −directions.

Example 155 Find the slope of the surface given by  ( ) =₋ 2 2 −  2₊25 8 5 2.5 0 -2.5 -5 5 2.5 0 -2.5 -5 0 -5 -10 -15 -20 -25 -30 x y z x y z

at the point ¡1₂ 1 2¢ _{in the −direction and in the −direction.} Example 156 Find the slopes of the surface given by

 ( ) = 1_{− ( − 1)}2_{− ( − 2)}2

at the point (1 2 1) in the −direction and in the −direction.

Partial derivative of a function of three or more variables:

If  =  (  )  A similar process is used to find the derivative of  with repect to  and with reapect to 

  = (  ) = lim∆→0  ( + ∆  )_{−  (  )} ∆   = (  ) = lim∆→0  (  + ∆ )_{−  (  )} ∆   = (  ) = lim∆→0  (   + ∆)_{−  (  )} ∆

In general, if  =  (1 2     )  there are  partial derivatives denoted

by

 

= (1 2     )   = 1 2     

Example 157 () To find the partial derivative of  (  ) = 33_2

− 4_2_{+ 7}5 _{with respect to   and }

() To find the partial derivative of  (  ) =  sin (2_{+ 2) with respect}

to   and 

() To find the partial derivative of  (   ) = ( +  + )  with respect to 