The derivative
|
|
We said in the last section that if we want to get
the instantaneous rate of change of the function, we will have to take the limit:
We call this limit the first derivative of the
function f. We
denote it by |
||
|
NB |
The first derivative is also a function of x. |
||
|
Definition |
The first
derivative is defined by the formula
|
||
|
|
This process of finding the
derivative of a function is called differentiation. |
||
|
NB |
If this limit does not exist
for some value of x, the given function will have no derivative for that value of x. We say that the function
is not differentiable at that point, or indifferentiable
at that point. |
||
|
NB |
The derivative is a limit. Therefore,
like other limits, it has a unique value. |
||
|
Example |
Let 1-
The change of y as x changes from 3 to 3+h. 2-
The average rate
of change of f in the same case. 3-
The instantaneous
rate of change when x=3. |
||
|
Example |
Find |
||
|
Now, let’s consider the curve of the function y=f(x). We call that line that
passes through two points on the curve a secant line. Suppose that the secant passes through the two
points
As h tends to zero, the point p tends to the point To find the slope
of the tangent, we can take the limit of the previous slope formula as h
tends to 0. So, the slope
|
|
||
|
|
Here we come to
this definition, which gives us another meaning of the derivative rather than
the instantaneous rate of change of a function. That is, the first derivative
of the function at a point also gives us the slope of the tangent to the
curve at that point. |
||
|
Definition |
The
tangent to the curve of y=f(x) at the point |
||
|
NB |
As we recall that
the equation of a line that passes through the point
Hence, the equation
of the tangent to the curve y=f(x) at the point
|
||
, 
, 
.
. Find:
.
. Then the 
. The secant line 
. The secant approaches what we call the tangent of the
curve at 
.
.
