Examples with Partial Derivatives                                           

 

 

 

 

 

 

 

 

1.

 

 

 

Example 1:  For the given function, f(x,y), find  fx(x,y),  fy(x,y),  fxy(x,y),  and fyx(x,y) 

 

                       f(x,y)  =  x2  exp (- y2  )

 

    ∂f/∂x  =   fx(x,y)  =  2 x  exp(-y2 )              (holding y constant)

 

    ∂f/∂y  =   fy(x,y) =  x2  (-2y) exp (- y2  )      (holding x constant)

 

 

    ∂/∂x[∂f/∂y]   =    fxy(x,y)  =     2 x exp(-y2 ) [-2y]     =  -4 x y exp (-y2 )

 

    ∂/∂y[∂f/∂x]   =   fyx(x,y)  =   (2x) [-2 y exp(-y2 )]   =   -4 x y exp (-y2 )

 

Note:  For continuous functions            fxy(x,y)    =       fyx(x,y) 

 

i. e.  The order of differentiation is irrelevant.

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

 

Example 2:  The concept of partial derivatives can be extended to a function of more than

two independent variables as shown in this example.

 

Given:   f(x, y, z)  =  cos (4x + 3y + 2z)      Find:    ∂3f/∂x∂y∂z   =     fxyz

 

Start with      ∂f/∂x  =    - 4 sin(4x + 3y + 2z)     

 

Then take the next derivative with respect to y giving   ∂2f/∂x∂y  =  -12 cos(4x + 3y + 2z)

 

Finally take the derivative with respect to  z  gives       ∂3f/∂x∂y∂z   

 

 

           ∂3f/∂x∂y∂z   =   24 sin(4x + 3y + 2z)      result

 

 

 

Note that the cosine function is continuous.  So the order of differentiation should be

immaterial.  Let us check this out by calculating    ∂3f/∂z∂y∂x

 

   Start with            ∂f/∂z  =    - 2 sin(4x + 3y + 2z)     

 

  Then               ∂2f/∂z∂y  =  -6 cos(4x + 3y + 2z)

 

And finally  ∂3f/∂x∂y∂z   =  24 sin(4x + 3y + 2z)      same result

 

Click here to continue with discussion on physical interpretation of the partial derivative.

 


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