Page 11 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation

Derivative of tan x
Let y = tan x
dy tan (x + δx) – tan x
= lim  4
dx x → 0 δx

= lim  sin (x + δx) – sin x 4 1
x → 0 cos (x + δx) cos x δx

= lim  sin (x + δx) cos x – cos (x + δx) sin x 4
x → 0 cos x cos (x + δx) · δx

sin [(x + δx) – x]

Using the formula
2 = x → 0 cos x cos (x + δx) · δx sin A cos B – cos A sin B = sin (A – B)
lim
sin δx
= x → 0 cos x cos (x + δx) · δx

lim
1 sin δx
= lim ·
x → 0 cos x cos (x + δx) δx
In the limit, as δx → 0, cos (x + δx) → cos x and sin δx → 1.
δx

d 1
2
Hence, dx (tan x) = cos x = sec x
2

Derivative of arcsin x (sin x)
–1
–1
Let y = sin x
x = sin y

Differentiating w.r.t. y, dx = cos y
dy
dy 1
∴ =
dx cos y
2
2
2
but cos y = 1 – sin y = 1 – x
dy 1
Hence, =
dx (1 – x )
2
d 1
–1
i.e. (sin x) =
dx (1 – x )
2
π
Note: y = sin x ⇒ – < y < π , since y is a principal value angle.
–1
2 2
For this range of values of y, cos x > 0
2
2
∴ cos x = (1 – x ) [not – (1 – x )]

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02 STPM Math(T) T2.indd 24 02/11/2018 12:43 PM