📚 DIFFERENTIATION

These properties help you handle complex functions easily by breaking them
down into simpler parts:

1. Linearity of Derivatives
 Addition Rule:
d
( f ( x ) + g ( x ) )=f ' ( x ) + g' ( x )
dx
 Subtraction Rule:
d
( f ( x )−g ( x ) ) =f ' ( x ) −g' ( x )
dx
 Constant Multiple Rule:
d
dx
[ c ⋅f ( x ) ]=c ⋅f ' ( x )

(where c is a constant)

2. Power Rule
If f ( x )=x n , then:
d n
( x )=n x n−1
dx
This is used in almost every basic polynomial derivative.

🔗 Chain Rule (Composite Function Rule)

Used when a function is inside another function:
If y=f ( g ( x )) , then:
dy df dg
= ⋅
dx dg dx
🔍 Example:

Let y=sin ( x2 )  Outer function: sin ( u ) , where

2
u=x
Apply chain rule:
 Derivative = cos ( x 2 ) ⋅2 x
 d
So, (sin ( x 2 ) )=2 x ⋅cos ( x2 )
dx

✖️Product Rule
Used when two functions are multiplied: If y=u ( x ) ⋅v ( x ), then:
d ' '
[u ⋅ v]=u ⋅v +u ⋅ v
dx
🔍 Example:
Let y=x 2 ⋅sin x
Apply product rule:
 2
u=x , v=sin x
 ' '
u =2 x , v =cos x
So,
dy 2
=2 x ⋅sin x+ x ⋅cos x
dx

➗ Quotient Rule
u(x)
Used when one function is divided by another: If y= , then:
v(x)
' '
dy u ⋅ v−u ⋅v
= 2
dx v
🔍 Example:
2
x
Let y=
sin x
 u=x , v=sin x
2

 u =2 x , v =cos x
' '

Then:
2
dy 2 x ⋅ sin x −x ⋅cos x
= 2
dx sin x
📌 Additional Properties
1. Derivatives of Constants
d
( c )=0
dx
2. Exponential Function
d x
( e ) =e x
dx
3. Logarithmic Differentiation
d 1
( log x )=
dx x

🧮 QUESTION BANK

Basic Level (Q1–Q10)

1. x 2
2. 3 x 3
3. x 5 + x3
4. 5 x−7
5. 2 x 2−4 x +1
6. x 3−3 x 2+5 x−7
7. 6
8. √x
1
9.
x
10. x−2

Intermediate Level (Q11–Q25)

2 3
11. x sin x 14. x ln x
12. cos x log x 15. sin x cos x
x 2x
13. e tan x 16. e
 log ( x 2 +1 )
x
17. 21. xe
1 22. ln ( √ x )
18. 2
x +1 23. ex
2

−1
19. tan x log 5 x
24.
−1
20. sin x x
25. x

Advanced Level (Q26–Q50)

x
26. x log x + e
27. ln ( x 3 +1 )
log x
28. x ⋅e
x x 39. x
2
29. tan x ⋅cot x 40. log x x

30. ln ( sin x ) 41. √ x ln x

31. log x +
2 1
x
42. ln ( 1−x
1+ x
)
32. ln ( tan x ) 43. sin x +cos x + tan x
33. ( x +1 )x 44. tan−1 ( e x )
2
34. x log 2 x 45. x
1/ x

35. x
e ⋅sin x 46. log 2 ( log 3 x )
x
36. x log x 47. ( ln x ) x
37. log ( log x ) 48. xx + xx
x

log x
38. e 49. log y=x log x
tan x
50. y=x
Q.N Q.N Q.N
Answer Answer Answer
o o o
−2 x x
1 2x 18 2 35 e ( sin x+ cos x )
( x 2+1 )
2 9x
2
19
1
1+ x
2 36 x x ( log x+1 ) 1+ ( 1
x log x+ x )
4 2
1 1
3 5 x +3 x 20 37
√1−x 2 x log x
x x
4 5 21 e +x e 38 1
1
5 4 x−4 22 39 x log x ( log x2 +1 )
2x
2 1−log x
6 2
3 x −6 x+ 5 23 2 x ex
2
40 ⋅
x ( log x )2
1 ln x 1
7 0 24 41 +
x ln 5 2 √x √ x
1 x 2
8 25 x ( log x+1 ) 42
2 √x 1−x
2

2 x 2
9 −1/ x 26 log x +1+e 43 cos x−sin x + sec x
2 x
−3 3x e
10 −2 x 27 3 44 2x
x +1 1+ e
1/ x log x−1
11 2 x sin x + x cos x
2
28 x x x x
x ( log x+1 ) e + x e 45 x ⋅
x
1
12 −sin x log x +cos x /x 29 2
sec x cot x−tan x csc x
2
46
x ln 3 ⋅log 3 x

13 x
e tan x +e sec x
x 2
30 cot x 47 ( ln x ) x ( lnxx +logln x )
2 1
−
x
2 2 x x x
14 3 x ln x + x 31 48 x ( log x+1 )+ x [log x ⋅ x ( log x+1 )]
x x2
x
15 2
cos x−sin x
2
32 2
sec x /tan x 49 x ( log x+1 )

16 2e
2x
33 x ( x +1 )
x−1
( log ( x +1 ) +1 ) 50 x tan x ( tanx x + sec x log x )
2

2x x
2
17 2 34 2 x log 2 x +
x +1 x ln 2