By now, the meaning and geometrical significance of differentiation should be pretty clear to you. We will use this knowledge to evaluate the derivatives of some standard functions in this section.
You will notice that while differentiating these functions, we will only use the expression for the ; we could equivalently use the also since all the functions we will be concerned with in this section are differentiable (except at discontinuous points); that the and are equal for each of these functions at a given can be easily verified.
(1)
This is intuitively true also since the graph for a constant function is a horizontal line.

(2)
This corresponds to the fact that the line is inclined at to the –axis (and is ).

(3)
This is again a straight forward result: ‘’ is the slope of so it must equal .

(4)
We have already obtained this result earlier.

(5)
[By the binomial expansion]
 
So, for example,
and and so on

(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
Since can be written as will be {because is a constant so it can be taken outside the differentiation operator; we will prove the validity of this step later}.
Therefore,
(16)
We use this relation now to simplify the numerator:
Notice that as
Also,
Therefore,
(17)
Notice that
Thus,
Note that for the last step, we have used the fact that differentiation operation is distributive over addition, i.e. . We will justify this later.
(18)
(19)
Now, it can easily be verified by rationalization that
Therefore,
(20)
Notice that
Therefore,
(21)
Notice again that
Therefore, as in the earlier cases,
It would be of help to you to get used to these differentiation on formulae as soon as possible, since they will be widely used subsequently.