Page 316 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

12.2 Definition of Successive Differentiation Notes

Consider,
A one variable function,
y = f(x) (x is independent variable and y depends on x.)
Here if we make any change in x there will be a related change in y.

dy
This change is called derivative of y w.r.t. x. denoted by f’(x) or y or y’ or called first order
1
dx
derivative of y w.r.t. x.
2
d y
f "( ) f '( ) ' y " y is called second order derivative of y w.r.t x.
x
x
2
dx 2
It gives rate of change in y w.r.t. rate of change in x.
1
Similarly,
3
d y
Third derivative of y is denoted by y 3 or "'( ) or or "' and
x
y
f
dx 3
So on…………………
(Above derivatives exist because, If y = f(x), then y = g(x), where g(x) is some function of x
1
depends on f(x) for e.g. if y = sinx then y = cosx, hence y = sinx and so on………)
1 2
Thus,
(n)
Derivatives of f(x) (or f) w.r.t. x are denoted by, f’(x), f”(x), f”’(x),……………f (x),………
Above process is called successive differentiation of f (x) w.r.t. x and f’, f”, f”’,………,f are called
(n)
successive derivatives of f.
th
f (x) denotes n derivative of f.
(n)
Notations:
Successive derivatives of y w.r.t. x are also denoted by,
1. y , y , y , ……….y ,……………. ………… or
1 2 3 n
2
n
3
dy d y d y d y
2. , , , ,……………… or
dx dx 2 dx 3 dx n
3. f’(x), f”(x), f”’(x), ………, f (x), ……… or
(n)
(n)
4. y’, y”, y”’, …………y , ………… or
5. Dy, D y, D y,…………., D y, …………….
n
2
3
d
Where D denotes .
dx
th
Value of n derivative of y = f(x) at x = a is denoted by,
n
d y
n
f (a), y (a), or
n n
dx
x a
n
(i.e. value can be obtained by just replacing x with a in f (x).)
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