
Differentiation 4-21
b. If f(x) = tan x, then prove that f ′(x) = sec
2
x.
To prove the above result, we need to know the following trigonometric identities:
sin (a + b) = sin a cos b + cos a sin b
cos (a + b) = cos a cos b - sin a sin b
tan
sin
cos
.x
x
x
=
Now,
f ' x
d
dx
x
x
x x x x
x
( )
sin
cos
cos cos s in sin
cos
=
=
⋅ − −
( )
2
=
+
= =
cos sin
cos
sec .
2 2
2
2
2
1
x x
x
x
Similarly, we can prove that f ′(x) = -cosec
2
x for f(x) = cot x.
c. If f(x) = sec x, then prove that f ′(x) = sec x tan x.
Now,
f ' x
d
dx x
x
x
x
x
x
(sec )
cos
[ sin ]
cos
sin
cos
sec ta
=
=
− −
= =
1
2
2
nn .x
Similarly, we can prove that f ′(x) = -cosec x cot x for f(x