Partial Derivatives

 


Consider   z=f(x,y) 

If we differentiate  z  w.r.t.  x   considering   y   as constant , we get the partial derivatives as

df/dx 

= dz/dx  or  fx  

Similarly keeping  x  constant we can differentiate  z  w.r.t.  y  we get ;

df/dy 

=dz/dy   or  fy   

If we differentiate   df/dx   again w.r.t.   x  keeping    y   as constant  

we get the second order partial derivatives 

(d/dx)(df/dx)

If we differentiate   df/dx  w.r.t.   y ,  keeping    x  as constant ,

we get another second order partial derivatives

(d/dy)(df/dx)

Similarly two more derivatives

(d/dx)(df/dy)

and (d/dy)(df/dy) .