Let 
be a function of 
the derivative or
Differential
coefficient of 
w.r.t. 
is denoted by
Where 
Similarly the second order derivative of y w.r.t. x is defined as
Similarly we get nth derivative or
Differential
coefficient of w.r.t. and is denoted
by
Application:
1. Series expansion of a function like Taylor’s and McLaurin's series expansion.
2. Solution of ordinary and partial differential equations.
Ex. Using Taylor’s Series
Example 1:
Find the nth derivative of 
Solution: differentiating y w.r.t. x we get,
…….......(1)
Differentiating again w.r.t. x, we get
…….......(2)
Similarly differentiating successively, we get
…….......(3)
.......(4)
… … … … … …
.......(5)
Example 2: Find the nth derivative of 
Solution: differentiating y w.r.t. x we get,
…….......(1)
Differentiating again w.r.t. x, we get
…….......(2)
Similarly differentiating successively, we get
…….......(3)
.......(4)
… … … … … …
.......(5)
Example 3: Find the nth derivative of 
Solution: differentiating y w.r.t. x we get,
…….......(1)
Differentiating again w.r.t. x, we get
....(2)
Similarly differentiating successively, we get
…….......(3)
….........(4)
… … … … … …
.......(5)
Example 4: Find the nth derivative of 
Solution: differentiating 
w.r.t. 
we get,
…….......(1)
Differentiating again w.r.t. , we get
…….......(2)
Similarly differentiating successively, we get
…….......(3)
Successively differentiating w.r.t. ,we get
…….......(4)
… … … … … … … …
The nth order derivative of y is given as
Example 5: Find the nth derivative of 
Solution: differentiating w.r.t. we get,
…….......(1)
Differentiating again w.r.t. , we get
…….......(2)
Similarly differentiating successively, we get
…….......(3)
Successively differentiating w.r.t. ,we get
…….......(4)
… … … … … … … …
The nth order derivative of y is given as
 |
Solution: differentiating w.r.t. 
we get,
.......(1)
Here we consider 
Thus we have 
and 
Now substituting these values in (1) we get,
 |
differentiating 
again w.r.t. we get,
.......(2)
Again, we consider
Thus we have and
Now substituting these values in (2) we get,
Thus we have
… … … … … … … … … … …
Example 7: Find the nth derivative of 
Solution: differentiating w.r.t. successively,
we get
… … … … … … … … … … …
Case- I
Case- II
Case-
III
Case-
IV
Case-
V
Example 8:
Find the nth derivative of 
Solution: differentiating w.r.t. we get,
Differentiating
again 
times w.r.t. , we get
 |
Example 9: Find the nth derivative of 
Solution: we have 
Let 
Putting 
we get
Putting 
we get
Thus 
Differentiating w.r.t. x n times we get,
For 
Compare with the formula
If 
then
Here 
, hence
Differentiating w.r.t. x n times we get,
For 
Compare with the formula
If then
Here 
, hence
Differentiating w.r.t. x n times we get,
