Page 10 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation
Differentiation of trigonometric functions

Derivative of sin x
Let y = sin x
dy sin (x + δx) – sin x
= lim
dx x → 0 δx
2 cos (x + δx) + x sin (x + δx) – x Using the formula

lim  2 2 4 sin A – sin B
=
A – B
A + B
x → 0 δx = 2 cos ––––– sin –––––
2
2
2 cos x + δx 2 sin δx
1

lim 2 2 2
=
x → 0 δx
δx
sin

lim cos x + δx 2
1
2
=
x → 0 2 δx

2
sin δx
In the limit, as δx → 0, cos x + δx 2 → cos x and δx 2 → 1
1
2
2
d
Hence dx (sin x) = cos x, where x is in radians.
Derivative of cos x
Let y = cos x
dy cos (x + δx) – cos x
= lim
dx x → 0 dx
Using the formula
–2 sin (x + δx) + x sin (x + δx) – x cos A – cos B

lim  2 2 4 = –2 sin ––––– sin –––––
A + B
A – B
=
x → 0 δx 2 2
–2 sin 1 x + δx 2 sin δx
= lim 2 2
x → 0 δx
sin δx
1
= lim –sin x + δx 2 2
x → 0 2 δx

2
sin δx
In the limit, as δx → 0, sin 1 x + δx 2 → sin x and 2 → 1
2
δx
2
d
Hence dx (cos x) = –sin x, where x is in radians.

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02 STPM Math(T) T2.indd 23 02/11/2018 12:43 PM