DERIVATIVIES

DERIVATIVIES:-
let f be a function defined on adomain containing anbhd of apoint
z_0 ,then the derivative of f at z_0 ,denoted by f ?(z_0 ) is defined by
f ?(z_0 )=lim?(z?z_0 )??(f(z)-f(z_0 ))/(z-z_0 )?
provided the limit exists .
f is said to be differentiable at z_0 if the derivative f ?(z_0 ) exists .
set ,?z=z-z_0
then f ?(z_0 )=lim?(?z?0)??(f(z_0+?z)-f(z_0 ))/?z?
equavalently,
f ?(z) =lim?(?z?0)??(f(z+?z)-f(z))/?z?
here |?z| is so small that f(z_0+?z)is defined.
*differentiation formulas:-
the definition of derivative
f ?(z)=lim?(?z?0)??(f(z+?z)-f(z))/?z?
is identical in form to that of the derivative of areal value function of areal
variable .
in fact ,the basic differentiation formulas given below can be derived from
that definition together with various theorems on limits ,by essentially
the same steps as the ones used in calculus.in these formulas.
the derivative of a function f at a point z is denoted by either
f ?(z) or d/dz f(z)
let c be a complex constant ,and let f be a funnction whose derivative
exists at a point z ,it is easy to show that .