Page 1
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .2
Question 1:
Find the derivative of
2
x - 2 at x = 10.
Solution 1:
Let f(x) =
2
x – 2. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
2
2
0
2 2 2
0
2
0
0
10 10
' 10 lim
10 2 10 2
lim
10 2.10. 2 10 2
lim
20
lim
lim 20 20 0 20
h
h
h
h
h
f h f
f
h
h
h
hh
h
hh
h
h
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ? ?
?
?
?
? ? ? ? ?
Thus, the derivative of
2
x – 2 at x = 10 is 20.
Question 2:
Find the derivative of 99x at x = 100.
Solution 2:
Let f(x) = 99x. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ?
0
0
0
0
0
100 100
' 100 lim
99 100 99 100
lim
99 100 99 99 100
lim
99
lim
lim 99 99
h
h
h
h
h
f h f
f
h
h
h
h
h
h
h
?
?
?
?
?
??
?
??
?
? ? ? ?
?
?
??
Thus, the derivative of 99x at x = 100 is 99.
Page 2
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .2
Question 1:
Find the derivative of
2
x - 2 at x = 10.
Solution 1:
Let f(x) =
2
x – 2. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
2
2
0
2 2 2
0
2
0
0
10 10
' 10 lim
10 2 10 2
lim
10 2.10. 2 10 2
lim
20
lim
lim 20 20 0 20
h
h
h
h
h
f h f
f
h
h
h
hh
h
hh
h
h
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ? ?
?
?
?
? ? ? ? ?
Thus, the derivative of
2
x – 2 at x = 10 is 20.
Question 2:
Find the derivative of 99x at x = 100.
Solution 2:
Let f(x) = 99x. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ?
0
0
0
0
0
100 100
' 100 lim
99 100 99 100
lim
99 100 99 99 100
lim
99
lim
lim 99 99
h
h
h
h
h
f h f
f
h
h
h
h
h
h
h
?
?
?
?
?
??
?
??
?
? ? ? ?
?
?
??
Thus, the derivative of 99x at x = 100 is 99.
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 3:
Find the derivative of x at x = 1.
Solution 3:
Let f(x) = x. Accordingly,
? ?
? ? ? ?
? ?
? ?
0
0
0
0
11
' 1 lim
11
lim
lim
lim 1
h
h
h
h
f h f
f
h
h
h
h
h
?
?
?
?
??
?
??
?
?
?
= 1
Thus, the derivative of x at x = 1 is 1.
Question 4:
Find the derivative of the following functions from first principle.
(i)
3
x – 27 (ii) (x – 1) (x – 2)
(ii)
2
1
x
(iv)
1
1
x
x
?
?
Solution 4:
(i) Let f(x) =
3
x – 27. Accordingly, from the first principle,
? ?
? ? ? ?
? ? ? ?
? ?
0
3
3
0
3 3 2 2 3
0
3 2 2
0
3 2 2
0
22
' lim
27 27
lim
33
lim
33
lim
lim 3 3
0 3 0 3
h
h
h
h
h
f x h f x
fx
h
x h x
h
x h x h xh x
h
h x h xh
h
h x h xh
xx
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ?
?
??
?
? ? ?
? ? ? ?
(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,
Page 3
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .2
Question 1:
Find the derivative of
2
x - 2 at x = 10.
Solution 1:
Let f(x) =
2
x – 2. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
2
2
0
2 2 2
0
2
0
0
10 10
' 10 lim
10 2 10 2
lim
10 2.10. 2 10 2
lim
20
lim
lim 20 20 0 20
h
h
h
h
h
f h f
f
h
h
h
hh
h
hh
h
h
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ? ?
?
?
?
? ? ? ? ?
Thus, the derivative of
2
x – 2 at x = 10 is 20.
Question 2:
Find the derivative of 99x at x = 100.
Solution 2:
Let f(x) = 99x. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ?
0
0
0
0
0
100 100
' 100 lim
99 100 99 100
lim
99 100 99 99 100
lim
99
lim
lim 99 99
h
h
h
h
h
f h f
f
h
h
h
h
h
h
h
?
?
?
?
?
??
?
??
?
? ? ? ?
?
?
??
Thus, the derivative of 99x at x = 100 is 99.
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 3:
Find the derivative of x at x = 1.
Solution 3:
Let f(x) = x. Accordingly,
? ?
? ? ? ?
? ?
? ?
0
0
0
0
11
' 1 lim
11
lim
lim
lim 1
h
h
h
h
f h f
f
h
h
h
h
h
?
?
?
?
??
?
??
?
?
?
= 1
Thus, the derivative of x at x = 1 is 1.
Question 4:
Find the derivative of the following functions from first principle.
(i)
3
x – 27 (ii) (x – 1) (x – 2)
(ii)
2
1
x
(iv)
1
1
x
x
?
?
Solution 4:
(i) Let f(x) =
3
x – 27. Accordingly, from the first principle,
? ?
? ? ? ?
? ? ? ?
? ?
0
3
3
0
3 3 2 2 3
0
3 2 2
0
3 2 2
0
22
' lim
27 27
lim
33
lim
33
lim
lim 3 3
0 3 0 3
h
h
h
h
h
f x h f x
fx
h
x h x
h
x h x h xh x
h
h x h xh
h
h x h xh
xx
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ?
?
??
?
? ? ?
? ? ? ?
(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ?
0
0
2 2 2
0
2
0
2
0
0
' lim
1 2 1 2
lim
2 2 2 2 2
lim
2
lim
23
lim
lim 2 3
23
h
h
h
h
h
h
f x h f x
fx
h
x h x h x x
h
x hx x hx h h x h x x x
h
hx hx h h h
h
hx h h
h
xh
x
?
?
?
?
?
?
??
?
? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ?
?
??
?
? ? ?
??
(iii) Let f(x) =
2
1
x
. Accordingly, from the first principle,
? ?
? ? ? ?
? ?
0
2 2
0
' lim
11
lim
h
h
f x h f x
fx
h
x
xh
h
?
?
??
?
?
?
?
? ?
? ?
2
2
2
2
0
1
lim
h
x x h
h
x x h
?
??
??
???
?
??
??
? ?
? ?
2
2
2
2
0
1
lim
h
x x h
h
x x h
?
??
??
???
?
??
??
? ?
222
2
2
0
12
lim
h
x x h hx
h
x x h
?
??
???
???
?
??
??
? ?
2
2
2
0
12
lim
h
h hx
h
x x h
?
??
??
? ??
?
??
??
? ?
2
2
2
0
2
lim
h
hx
x x h
?
??
??
???
?
??
??
? ?
2 3
2
0 2 2
0
x
x
xx
??
??
?
(iv) Let f(x) =
1
1
x
x
?
?
. Accordingly, from the first principle,
Page 4
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .2
Question 1:
Find the derivative of
2
x - 2 at x = 10.
Solution 1:
Let f(x) =
2
x – 2. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
2
2
0
2 2 2
0
2
0
0
10 10
' 10 lim
10 2 10 2
lim
10 2.10. 2 10 2
lim
20
lim
lim 20 20 0 20
h
h
h
h
h
f h f
f
h
h
h
hh
h
hh
h
h
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ? ?
?
?
?
? ? ? ? ?
Thus, the derivative of
2
x – 2 at x = 10 is 20.
Question 2:
Find the derivative of 99x at x = 100.
Solution 2:
Let f(x) = 99x. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ?
0
0
0
0
0
100 100
' 100 lim
99 100 99 100
lim
99 100 99 99 100
lim
99
lim
lim 99 99
h
h
h
h
h
f h f
f
h
h
h
h
h
h
h
?
?
?
?
?
??
?
??
?
? ? ? ?
?
?
??
Thus, the derivative of 99x at x = 100 is 99.
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 3:
Find the derivative of x at x = 1.
Solution 3:
Let f(x) = x. Accordingly,
? ?
? ? ? ?
? ?
? ?
0
0
0
0
11
' 1 lim
11
lim
lim
lim 1
h
h
h
h
f h f
f
h
h
h
h
h
?
?
?
?
??
?
??
?
?
?
= 1
Thus, the derivative of x at x = 1 is 1.
Question 4:
Find the derivative of the following functions from first principle.
(i)
3
x – 27 (ii) (x – 1) (x – 2)
(ii)
2
1
x
(iv)
1
1
x
x
?
?
Solution 4:
(i) Let f(x) =
3
x – 27. Accordingly, from the first principle,
? ?
? ? ? ?
? ? ? ?
? ?
0
3
3
0
3 3 2 2 3
0
3 2 2
0
3 2 2
0
22
' lim
27 27
lim
33
lim
33
lim
lim 3 3
0 3 0 3
h
h
h
h
h
f x h f x
fx
h
x h x
h
x h x h xh x
h
h x h xh
h
h x h xh
xx
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ?
?
??
?
? ? ?
? ? ? ?
(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ?
0
0
2 2 2
0
2
0
2
0
0
' lim
1 2 1 2
lim
2 2 2 2 2
lim
2
lim
23
lim
lim 2 3
23
h
h
h
h
h
h
f x h f x
fx
h
x h x h x x
h
x hx x hx h h x h x x x
h
hx hx h h h
h
hx h h
h
xh
x
?
?
?
?
?
?
??
?
? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ?
?
??
?
? ? ?
??
(iii) Let f(x) =
2
1
x
. Accordingly, from the first principle,
? ?
? ? ? ?
? ?
0
2 2
0
' lim
11
lim
h
h
f x h f x
fx
h
x
xh
h
?
?
??
?
?
?
?
? ?
? ?
2
2
2
2
0
1
lim
h
x x h
h
x x h
?
??
??
???
?
??
??
? ?
? ?
2
2
2
2
0
1
lim
h
x x h
h
x x h
?
??
??
???
?
??
??
? ?
222
2
2
0
12
lim
h
x x h hx
h
x x h
?
??
???
???
?
??
??
? ?
2
2
2
0
12
lim
h
h hx
h
x x h
?
??
??
? ??
?
??
??
? ?
2
2
2
0
2
lim
h
hx
x x h
?
??
??
???
?
??
??
? ?
2 3
2
0 2 2
0
x
x
xx
??
??
?
(iv) Let f(x) =
1
1
x
x
?
?
. Accordingly, from the first principle,
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
0
0
22
0
0
0
' lim
11
11
lim
1 1 1 1
1
lim
11
11
1
lim
11
12
lim
11
2
lim
11
2
11
h
h
h
h
h
h
f x h f x
fx
h
x h x
x h x
h
x x h x x h
h x x h
x hx x x h x hx x x h
h x x h
h
h x x h
x x h
xx
?
?
?
?
?
?
??
?
? ? ? ??
?
??
? ? ?
??
?
?? ? ? ? ? ? ? ?
?
??
? ? ?
??
??
? ? ? ? ? ? ? ? ? ? ?
??
? ? ?
??
??
??
?
?
??
? ? ?
??
??
?
?
??
? ? ?
??
??
??
??
? ?
2
2
1 x ?
Question 5:
For the function
F(x) =
100 99 2
... 1
100 99 2
x x x
x ? ? ? ? ?
Prove that f’(1) = 100 f’(0)
Solution 5:
The given function is
F(x) =
100 99 2
... 1
100 99 2
x x x
x ? ? ? ? ?
? ? ? ?
? ?
100 99 2
100 99 2
1
( ) ... 1
100 99 2
( ) ... 1
100 99 2
On using theorem ,we obtain
nn
d d x x x
f x x
dx dx
d d x d x d x d d
f x x
dx dx dx dx dx dx
d
x nx
dx
?
??
? ? ? ? ? ?
??
??
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
?
99 98
100 99 2
( ) ... 1 0
100 99 2
d x x x
fx
dx
? ? ? ? ? ?
Page 5
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Exercise 2 1 .2
Question 1:
Find the derivative of
2
x - 2 at x = 10.
Solution 1:
Let f(x) =
2
x – 2. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
2
2
0
2 2 2
0
2
0
0
10 10
' 10 lim
10 2 10 2
lim
10 2.10. 2 10 2
lim
20
lim
lim 20 20 0 20
h
h
h
h
h
f h f
f
h
h
h
hh
h
hh
h
h
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ? ?
?
?
?
? ? ? ? ?
Thus, the derivative of
2
x – 2 at x = 10 is 20.
Question 2:
Find the derivative of 99x at x = 100.
Solution 2:
Let f(x) = 99x. Accordingly,
? ?
? ? ? ?
? ? ? ?
? ?
0
0
0
0
0
100 100
' 100 lim
99 100 99 100
lim
99 100 99 99 100
lim
99
lim
lim 99 99
h
h
h
h
h
f h f
f
h
h
h
h
h
h
h
?
?
?
?
?
??
?
??
?
? ? ? ?
?
?
??
Thus, the derivative of 99x at x = 100 is 99.
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
Question 3:
Find the derivative of x at x = 1.
Solution 3:
Let f(x) = x. Accordingly,
? ?
? ? ? ?
? ?
? ?
0
0
0
0
11
' 1 lim
11
lim
lim
lim 1
h
h
h
h
f h f
f
h
h
h
h
h
?
?
?
?
??
?
??
?
?
?
= 1
Thus, the derivative of x at x = 1 is 1.
Question 4:
Find the derivative of the following functions from first principle.
(i)
3
x – 27 (ii) (x – 1) (x – 2)
(ii)
2
1
x
(iv)
1
1
x
x
?
?
Solution 4:
(i) Let f(x) =
3
x – 27. Accordingly, from the first principle,
? ?
? ? ? ?
? ? ? ?
? ?
0
3
3
0
3 3 2 2 3
0
3 2 2
0
3 2 2
0
22
' lim
27 27
lim
33
lim
33
lim
lim 3 3
0 3 0 3
h
h
h
h
h
f x h f x
fx
h
x h x
h
x h x h xh x
h
h x h xh
h
h x h xh
xx
?
?
?
?
?
??
?
??
? ? ? ?
??
?
? ? ? ?
?
??
?
? ? ?
? ? ? ?
(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ?
? ?
0
0
2 2 2
0
2
0
2
0
0
' lim
1 2 1 2
lim
2 2 2 2 2
lim
2
lim
23
lim
lim 2 3
23
h
h
h
h
h
h
f x h f x
fx
h
x h x h x x
h
x hx x hx h h x h x x x
h
hx hx h h h
h
hx h h
h
xh
x
?
?
?
?
?
?
??
?
? ? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ?
?
??
?
? ? ?
??
(iii) Let f(x) =
2
1
x
. Accordingly, from the first principle,
? ?
? ? ? ?
? ?
0
2 2
0
' lim
11
lim
h
h
f x h f x
fx
h
x
xh
h
?
?
??
?
?
?
?
? ?
? ?
2
2
2
2
0
1
lim
h
x x h
h
x x h
?
??
??
???
?
??
??
? ?
? ?
2
2
2
2
0
1
lim
h
x x h
h
x x h
?
??
??
???
?
??
??
? ?
222
2
2
0
12
lim
h
x x h hx
h
x x h
?
??
???
???
?
??
??
? ?
2
2
2
0
12
lim
h
h hx
h
x x h
?
??
??
? ??
?
??
??
? ?
2
2
2
0
2
lim
h
hx
x x h
?
??
??
???
?
??
??
? ?
2 3
2
0 2 2
0
x
x
xx
??
??
?
(iv) Let f(x) =
1
1
x
x
?
?
. Accordingly, from the first principle,
Chapter 2 1 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
0
0
0
22
0
0
0
' lim
11
11
lim
1 1 1 1
1
lim
11
11
1
lim
11
12
lim
11
2
lim
11
2
11
h
h
h
h
h
h
f x h f x
fx
h
x h x
x h x
h
x x h x x h
h x x h
x hx x x h x hx x x h
h x x h
h
h x x h
x x h
xx
?
?
?
?
?
?
??
?
? ? ? ??
?
??
? ? ?
??
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?? ? ? ? ? ? ? ?
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??
? ? ?
??
??
? ? ? ? ? ? ? ? ? ? ?
??
? ? ?
??
??
??
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??
? ? ?
??
??
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??
? ? ?
??
??
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??
? ?
2
2
1 x ?
Question 5:
For the function
F(x) =
100 99 2
... 1
100 99 2
x x x
x ? ? ? ? ?
Prove that f’(1) = 100 f’(0)
Solution 5:
The given function is
F(x) =
100 99 2
... 1
100 99 2
x x x
x ? ? ? ? ?
? ? ? ?
? ?
100 99 2
100 99 2
1
( ) ... 1
100 99 2
( ) ... 1
100 99 2
On using theorem ,we obtain
nn
d d x x x
f x x
dx dx
d d x d x d x d d
f x x
dx dx dx dx dx dx
d
x nx
dx
?
??
? ? ? ? ? ?
??
??
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
?
99 98
100 99 2
( ) ... 1 0
100 99 2
d x x x
fx
dx
? ? ? ? ? ?
Chapter 13 – Limits and Derivatives Maths
______________________________________________________________________________
? ?
? ?
? ? ? ?
? ? ? ?
99 98
99 98
99 98
100
... 1
' ... 1
At x =0,
' 0 1
At x =1,
' 1 1 1 ... 1 1 1 1 ... 1 1 1 100 100
Thus, ' 1 100 ' 0
terms
x x x
f x x x x
f
f
ff
? ? ? ? ?
? ? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ?
??
Question 6:
Find the derivative of
1 2 2 1
...
n n n n n
x ax a x a x a
? ? ?
? ? ? ? ? for some fixed real number a.
Solution 6:
Let f(x) =
1 2 2 1
...
n n n n n
x ax a x a x a
? ? ?
? ? ? ? ?
? ?
? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ? ? ?
? ? ? ?
1 2 2 1
1 2 2 1
1
1 2 2 3 1
1 2 2 3 1
'( ) ...
... 1
On using theorem ,we obtain
'( ) 1 2 ... 0
1 2 ...
n n n n n
n n n n n
nn
n n n n n
n n n n
d
f x x ax a x a x a
dx
d d d d d
x a x a x a x a
dx dx dx dx dx
d
x nx
dx
f x nx a n x a n x a a
nx a n x a n x a
? ? ?
? ? ?
?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ?
? ? ? ? ? ?
?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
Question 7:
For some constants a and b, find the derivative of
(i) (x – a) (x – b) (ii) (ax2 + b)2 (iii)
xa
xb
?
?
Solution 7:
(i) Let f (x) = (x – a) (x – b)
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