Page 308 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

dy Notes
If f x is a differentiable function, then by differentiating it w.r.t. x, we get
dx
2
d y
f x ...(ii)
dx 2
Similarly by differentiating it w.r.t. x, we get
3
d y
f x ...(iii)
dx 3
Again by differentiating it w.r.t. x, we get

4
d y
f iv x ...(iv)
dx 4
and so on.
dy
This process of finding higher ordered derivatives is called successive differentiation. is
dx
3
2
d y d y
called first derivative, 2 is called the second derivative, 3 is called the third derivative
dx dx
4
d y
and is called fourth derivative and so on.
dx 4
n
d y d n 1 y
th
In general, n is called the n derivative, which is obtained by differentiating n 1 w.r.t. x.
dx dx
d n y d n
The n derivative of y = f(x) is denoted by the symbols y n , f (n ) (x ), , [ f (x )]
th
dx n dx n
Examples: Find the second, third, fourth derivatives of the following functions:

1 ax b
3
2
4
1. x 5x 7x 2x 2. ax 2 bx c 3.
x cx d
4. x log x 5. xe x
Solution:
1
3
2
4
1. Let y x 5x 7x 2x
x
dy 3 2 1
4x 15x 14x 2 ]
dx x 2
2
d y 2 2
12x 30x 14
dx 2 x 3
d 3 y 6
24x 30
dx 3 x 4
d 4 y 24
24
dx 4 x 5

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