Page 7 - PRE-U STPM MATHEMATICS (T) TERM 2

Mathematics Term 2 STPM Chapter 2 Differentiation
d (x + x) – x 3
3
x = lim
3
dx x → 0 x
2
2
= lim [3x + 3x x + (x) ]
x → 0
= 3x 2
3 – 1
= 3x
d (x + x) – x –1
–1
–1
x = lim
dx x → 0 x
= lim – 1
x → 0 x(x + x)
1
2 = – x 2
= –x –2
= –1(x –1 – 1 )
Hence,
n
d x = nx n – 1 , n  R.
dx
Example 3

Differentiate with respect to x.
(a) x –3 (b)   x
(c) –5x 4 (d) 3
x 6
Solution: (a) Let y = x –3
dy 3
–4
= –3x –3 – 1 = –3x = –
dx x 4
(b) Let y =   x = x — 1 2
dy = 1 x — – 1 = 1 x = 1
1
1
2
–—
2
dx 2 2 2  x
(c) Let y = –5x 4
dy = –5(4x 4 – 1 ) = –20x 3
dx
(d) Let y = 3 6 = 3x – 6
x
dy = 3[–6x –6 – 1 ] = –18x = – 18
–7

dx x 7
Exercise 2.2

Find the derivatives of the following functions with respect to x.

5
1. 5 2. 3x 3. –4x 4. x 5. x –4
1
—
x
3
6. x 7. x 8.  9. 1 10. 3 x 2
–1
3
  x
3
10
–3
11. 5x 12. 7x 13. 5 14. 7 x 15. –2
4x 4 8 x 8
20
02 STPM Math(T) T2.indd 20 02/11/2018 12:43 PM