Page 33 - Calculus with Complex Numbers

D e r i v at i ve s

3 . I D i ffe re nti ab i I i ty and co nti n u i ty
For a real function f @) of a real variable .x the derivative ./'/ @) is delined as the
limit
? f ( f (x + h) - flx)
h
yt)
. 1 ) = li h .
...
Observe that (see Figtlre 3.1)
f(x + h) - flx)
h
is the gradient of the line P Q which converges to the tangent at P as Q --> P . So
f' @) is the gradient of the tangent at P.
For example, if flx) = .x2 then we have

f(x V h) - flx) V V h)l - .12 .x2 + lxh + hl - .x2
h h h
lxh + hl
= = 2.x + h,
h

Ffgure .? . /