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NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

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 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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??
? ? ?
??
??
??
??
? ?
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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FAQs on NCERT Solutions Class 11 Maths Chapter 12 - Limits and Derivatives

1. What is the definition of a limit in calculus?
Ans. In calculus, a limit is the value that a function approaches as the input (or independent variable) approaches a certain value. It represents the behavior of a function near a particular point.
2. How is the limit of a function calculated?
Ans. The limit of a function can be calculated by evaluating the function as the input approaches a specific value. If the function approaches a single value as the input gets closer to the specified value, that value is the limit.
3. What is the difference between a left-hand limit and a right-hand limit?
Ans. A left-hand limit is the value that a function approaches as the input approaches a specific value from the left side, while a right-hand limit is the value that a function approaches as the input approaches the same value from the right side.
4. How are derivatives related to limits in calculus?
Ans. Derivatives in calculus are closely related to limits, as the derivative of a function at a particular point is defined as the limit of the average rate of change of the function as the interval around the point shrinks to zero.
5. Can limits be used to find the slope of a curve at a specific point?
Ans. Yes, limits can be used to find the slope of a curve at a specific point by calculating the derivative of the function at that point. The derivative represents the slope of the tangent line to the curve at that particular point.
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