Page 12 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation
–1
Derivative of arc cos x (cos x)
–1
Let y = cos x
x = cos y
dx
= –sin y
dy
dy 1
= –
dx sin y
2
2
but sin y = 1 – cos y
= 1 – x 2
dy 1
∴ = –
dx (1 – x )
2
2
d 1
–1
Hence, (cos x) = –
dx (1 – x )
2
–1
Derivative of arc tan x (tan x)
Let y = tan x
–1
x = tan y
2
dx = sec y
dy
dy 1
=
2
dx sec y
= 1
2
1 + tan y
= 1
1 + x 2

–1
Hence, d (tan x) = 1
dx 1 + x 2

Rules of differentiation

Differentiation of sums and differences of functions
Consider two functions of x, p(x) and q(x), and let f(x) = p(x) + q(x).
From the derived definition,
d [p(x + x) + q(x + x)] – [p(x) + q(x)]
[f(x)] = lim
dx x → 0 x
[p(x + x) – p(x)] + [q(x + x) – q(x)]
= lim
x → 0 x
p(x + x) – p(x) q(x + x) – q(x)
= lim + lim
x → 0 x x → 0 x
d d
= [p(x)] + [q(x)]
dx dx

d d d
Hence, [f(x)] = [p(x)] + [q(x)]
dx dx dx

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02 STPM Math(T) T2.indd 25 02/11/2018 12:43 PM