Page 7 - Math SL HB Sem 2

The Dcrivative Functiol

Consider a general function y =/(:,) wherc Ais (a fft)) andB is (x+la fft+h)).
v
!=fu

1tu+h'l

k)

z 2 t

f(x+h)- f(x)
The chord AB has slope =
x+h-x

f(x+h)-f(x)
h

If we now let B move closer to A, the slope ofAB approaches the slope of the tangent
at A.

So, the slope of the tangent at the variable point (>rq (x)) is the limiting value of
f(x+h)-f(x)
as h approaches O.
h

Since this slope contains the variable x it is called a slope function.

DERIVATIVE FTJNCTION

The slope function, also known as the deriyed fuuction, or derivative
function or simply the derivative is defined as
Iim f(x+h)- f (x)
f(x)=
h--+0 h
This formal definition can be used to diflerentiate any function. This process is called
differentiating from fi rst principles.

Notation

There are many ways to denote the derivative ofa function y = f(x). Besides f (r the
)
most common notations are these:
,.*.**,,a, n..r(,)

:fr
t f ' i. *r" f^t (first order) derivaive of y wilt respect to r )