The Basic Rules of Derivative are described along with the examples and MCQs. From the Constant rule to the Chian Rule.

1. RULES
OF DERIVATIVES

2. Derivative
The instantaneous Rate of Change of a
function.
Application
Derivatives are met in many engineering
and science problems, especially, when
modelling the behavior of moving
objects. Like in;
Gaming , designing, etc.

3. Rules.
THE
CONSTANT
RULE
THE
CONSTANT
MULTIPLE
RULE
THE
ADDITION
AND
SUBTRACTIO
N RULE
THE POWER
RULE
THE
PRODUCT
RULE
THE
QUOTIENT
RULE
THE CHAIN
RULE

4. CONSTANT | RULE
The derivative of a constant function f(x) = c is 0. That is, f’(x) = 0. This can also be
written as :
𝑑𝑓
𝑑𝑥
=
𝑑
𝑑𝑥
(c) = 0.
Examples:
o If f(x) =
1
2
then f’(0) = 0.
oIf y = −π, then
𝑑𝑦
𝑑𝑥
=
𝑑
𝑑𝑥
(-π) = 0.

5. CONSTANT
MULTIPLE
| RULE
If c is any constant, then
𝑑
𝑑𝑥
(cf(x)) = cf’(x).
Example:
If y = 2x then,
𝑑
𝑑𝑥
2𝑥 = 2
𝑑
𝑑𝑥
𝑥 = 2(1) .
f’(x) = 2.

6. Addition and Subtraction |Rule
If f and g are differentiable functions, then
Addition Rule
𝑑
𝑑𝑥
𝑓 + 𝑔 =
𝑑
𝑑𝑥
𝑓 +
𝑑
𝑑𝑥
𝑔
Subtraction Rule
𝑑
𝑑𝑥
𝑓 − 𝑔 =
𝑑
𝑑𝑥
𝑓 −
𝑑
𝑑𝑥
𝑔

7. Addition and Subtraction |Rule
Examples:
oIf y = 3x + 5, then
𝑑𝑦
𝑑𝑥
=
𝑑
𝑑𝑥
(3x) +
𝑑
𝑑𝑥
(5) = 3(1) + 0 = 3.
oIf y = 70 – x, then
𝑑𝑦
𝑑𝑥
=
𝑑
𝑑𝑥
(70) -
𝑑
𝑑𝑥
(x) = 0 - 1 = -1.

8. PRODUCT |
RULE
If f and g are differentiable functions, then
𝑑
𝑑𝑥
𝑓 × 𝑔 = 𝑓
𝑑
𝑑𝑥
𝑔 + 𝑔
𝑑
𝑑𝑥
𝑓 .
Example:
If f(x) = 𝑥2
𝑒 𝑥
, then
𝑥2
𝑑
𝑑𝑥
𝑒 𝑥 + 𝑒 𝑥
𝑑
𝑑𝑥
𝑥2 .
𝑥2(𝑒 𝑥) + 𝑒 𝑥(2x).

9. Power | RULE
If n is any number as a power, then
𝑑
𝑑𝑥
(𝑥 𝑛 ) = n (𝑥 𝑛−1 )
𝑑
𝑑𝑥
(x) .
Examples:
oIf f(x) = 𝑥4 then f’(x) = 4𝑥3(1)= 4 𝑥3 .
o If f(x) = 𝑥−45 then f’(x) = -45𝑥−46(1)= -45 𝑥−46 .

10. QUOTIENT|
RULE
If f and g are differentiable functions, then
𝑑
𝑑𝑥
(
𝑓
𝑔
) =
𝑔
𝑑
𝑑𝑥
𝑓 −𝑓
𝑑
𝑑𝑥
(𝑔)
𝑔 ^2
Example:
If
𝑑
𝑑𝑥
(
𝑥2
𝑥+5
),
(𝑥+5)
𝑑
𝑑𝑥
𝑥2 −𝑥2 𝑑
𝑑𝑥
(𝑥+5)
(𝑥+5)^2
F’(x)=
2𝑥 −𝑥2
(𝑥+5)
=
𝑥(2−𝑥)
(𝑥+5)

11. The Chain Rule
If f(u) is differentiable at the point
u = g(x) and g(x)
is differentiable at x, then the
composite function:
[f (g(x))] = f’(g(x))g’(x).

14. 𝒅
𝒅𝒙
𝟏
𝒙
= ?
a- 𝒙 𝟐
b- − 𝒙 𝟐
c-
𝟏
𝒙 𝟐
d- -
𝟏
𝒙 𝟐
Q:1

15. y= 5𝑒3𝑥
+sinx, then,
𝑑𝑦
𝑑𝑥
=?
a- 5𝑒3𝑥
+ cosx
b- 15𝑒3𝑥 + cosx
c- 15𝑒3𝑥 − cosx
d- 2.66𝑒3𝑥 - cosx
Q:2

16. y=
𝑥+𝑎
𝑠𝑖𝑛𝑎
, then,
𝑑𝑦
𝑑𝑥
=?
a- -
𝟏
𝒔𝒊𝒏𝒂
b- sina –cosx
c- sina –x-a
d-
𝟏
𝒔𝒊𝒏𝒂
Q:3

17. x=2t and y=t^2, then,
𝑑𝑦
𝑑𝑥
=?
a- 4𝑡
b- 2
c- 𝑡
d- 4
Q:4

18. Y=
sin 𝑎
cos 𝑎
then,
𝑑𝑦
𝑑𝑥
=?
a- sec^2(ax + b)
b-
cos 𝑎
sin 𝑎
c- -
cos 𝑎
sin 𝑎
d- 0
Q:5

19. “Mathematics
is like love,
a
simple idea
but,
it can get
complicated.”