Page 6 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation
Differentiation of standard functions

Derivative of a constant
Consider the function y = c, where c is a constant. y
dy f(x + x) – f(x)
From the derived definition = lim
dx x → 0 x
As f(x + x) = c = f(x), c y = c
∴ f(x + x) – f(x) = c – c
x x
= 0 0 x
dy 2
Hence, = lim 0 Figure 2.2
dx x → 0
= 0

Geometrically, y = c represents a straight line parallel to the x-axis.

Therefore, its gradient is zero.
d (c) = 0, where c is a constant
dx

Derivative of af(x) where a is a constant
From the derived definition,
d a f(x + x) – a f(x)
[a f(x)] = lim
dx x → 0 x

= lim  a[f(x + x) – f(x)] 4
x → 0 x
= lim a  lim f(x + x) – f(x)] 4
x → 0 x → 0 x
d
= a [f(x)]
dx

d d
Hence, [a f(x)] = a [f(x)]
dx dx

Derivative of x n

Consider the following differentiation from first principles.
d (x + x) – x 2
2
x = lim
2
dx x → 0 x
= lim (2x + x)
x → 0
= 2x
= 2x 2 – 1

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02 STPM Math(T) T2.indd 19 02/11/2018 12:43 PM