The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists Derivative of f at c is denoted by f ′(c) or 
( ( )) |c
d
dxf x The
function defined by
0
(
)
( )
( )
lim
h
f x
h
f x
f
x
h
→
+
−
′
=
wherever the limit exists is defined to be the derivative of f The derivative of f is
denoted by f ′(x) or 
d( ( ))
dxf x
or if y = f (x) by dy
dx or y′