Derivatives

What is Derivatives?

The derivation of the word derivation is the Latin derivationem, meaning "a leading off or turning away
from."

in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental
to the solution of problems in calculus and differential equations.

In calculus, "deriving," or taking the derivative, means to find the "slope" of a given function.
Derivatives, on the other hand, are a measure of the rate of change, but they apply to almost any
function.

The Derivative as a Slope of the tangent line

A tangent line to the function f(x) at the point x=a is a line that just touches the graph of the function at
the point in question and is “parallel” (in some way) to the graph at that point.

Definition of the Derivative

Δy
The derivative of y with respect to x is the limit of the ratio when Δx approaches zero.
Δx
dy
The derivative is designated by the symbol :
dx
dy Δy f ( x +∆ x )−f (x )
= lim = lim
dx ∆ x→ 0 Δ x ∆ x→ 0 ∆x
' ' d
Other symbols for derivative are y , f ( x ) , D z y , f ( x) .
dx
Note: Since the symbol y ' does not explicitly indicate the independent variable, this notation should be
used only when no confusion or ambiguity can arise.

The operation of finding the derivative is called differentiation.

Determination of the Derivative

To obtain the derivative of any function, the general process is as follow:

1. Replace.
- Replace x by x +∆ x , and y by y + ∆ y :
(1) y=f ( x ) ,
(2) y + ∆ y=f (x+ ∆ x)
2. Subtraction
- By subtraction, eliminate y between (1) and (2), thus obtaining a formula for ∆ y in terms of
x and ∆ y
(3) ∆ y =f ( x +∆ x )−f ( x )
3. Throw.
 - By some suitable transformation, throw the right member of (3) into a form which contains
∆ x explicitly as a factor.
4. Divide
- Divide through by ∆ x :
∆ y f ( x+ ∆ x )−f ( x)
=
∆x ∆x
5. Determine.
- Determine the limit as ∆ x approaches zero.

Example 1:

Differentiate the example y=x 3−2 x

1. y + ∆ y=( x+ ∆ x )3−2 ( x+ ∆ x ) .
2. ∆ y =( x+ ∆ x )3−2 ( x+ ∆ x )−x 3+ 2 x
3. ¿ x 3+ 3 x 3 ∆ x +3 x ( ∆ x )2 + ( ∆ x )3−2 x−2 ∆ x−x 3 +2 x
3 2 3
¿ 3 x ∆ x +3 x ( ∆ x ) + ( ∆ x ) −2 ∆ x
3 2 3
Δ y 3 x ∆ x +3 x ( ∆ x ) + ( ∆ x ) −2 ∆ x
4. =
Δx ∆x
2 2
∆ x (3 x +3 x ∆ x+ ( ∆ x ) −2)
¿
∆x
5. y =3 x + 3 x (0)+ ( 0 )2−2
' 2

' 2
y =3 x −2
Example 2:

1
Find the derivative of the function x=
t
1
1. x +∆ x=
t+ ∆ t
1 1
2. ∆ x= −
t +∆ t t
1−(t+ ∆t ) −∆ t
3. ¿ =
(t +∆ t)t (t+ ∆ t)t
Δx −1
4. =
Δ t ( t +∆ t ) t
dx −1
5. =
d t ( t+ 0 ) t
d x −1
= 2
dt t

Differentiation Rules
Constant Rule
 d
- Let c be a constant. If f(x)=c, then f’(x)=0. Or can be written as c=0
dx
Example:
1. y=4 (since 4 is a constant, its derivative is o
y’=0
2. f(x)=10
f’(x)=0

Power Rule

- If n is any real number, then:

d n
[ x ]=n x n−1
dx
Example:
1. x 3
d 3
[ x ]=3 x 2−1
dx
1
¿2 x
¿2 x
2. x 12
d 12
[ x ]=12 x 1 2−1
dx
11
¿ 12 x
Constant multiple rule

- The derivative of constant times a function is the constant times the derivative of the
function. If c is a constant and f is a differentiable function, then,

( dxd ) cf ( x )=c ( dxd ) f ( x )

Example 1.

Derivative of a function ( dxd ) 3 x 4

( dxd ) 3 x =3( dxd ) x

4 4

¿ 3( ) 4 x
d 4−1
dx
3
¿ ( 3) 4 x
3
¿ 12 x
Example 2.

Derivative of a function ( dxd )−x

( dxd )−x =( dxd )(−1) x
 ¿(−1) ( dxd ) x
¿ ¿1)1
¿−¿1
Sum and difference rule
If f ( x )=u ( x) ± v ( x )then,
d d
f ' ( x )= [u ( x ) ]± [v ( x ) ] as long as u' ( x ) and v ' ( x )exist
dx dx
Example 1.
Find the derivative of g ( x )=11 x 2+ 4 x−2
2
g ( x )=11 x + 4 x−2
d d d
g' ( x )= ( 11 x ) + ( 4 x ) − ( 2 )
2
dx dx dx
d 2 d d
¿ 11∙ ( x ) + 4 ∙ ( x ) − ( 2 )
dx dx dx
¿ 11 ( 2 x ) + 4 ( 1 x ) −0
2−1 1−1

¿ 22 x+ 4
Product and Quotient Rule

The following guidelines must be followed if f and g are two distinct differentiable functions of x.

d
dx
[ f ( x ) g ( x ) ]=f ( x ) g' ( x )+ g ( x ) f '(x )

[ ]
' '
d f ( x ) g ( x ) f ( x )−f ( x ) g ( x )
=
dx g ( x ) [g ( x )]
2

Example 1.
2
y=( x )(2 x +3)
Let u=x2

du=x
Let v=2 x+3

dv =2
'
y =udv + vdu

y ' =( x 2 ) ( 2 ) + ( 2 x +3 )( x )

¿ ( 2 x 2) + ( 2 x 2 +3 x )
2 2
¿ 2 x +2 x + 3 x
 2
¿ 4 x +3 x
Example 2.

3 x−4
f ( x )= 2
2 x +5
Let u=3 x −4

du=3
Let v=2 x2 +5

dv =4 x
' vdu−udv
f ( x)= 2
v

( 2 x 2+ 5 ) ( 3 )−(3 x −4)( 4 x )
f ' ( x )= 2
( 2 x 2 +5 )
( 6 x 2 +1 5 )−(12 x 2−16 x )
f ' ( x )= 2
( 2 x 2 +5 )
2 2
6 x +15−12 x +16 x
f ' ( x )= 2
( 2 x 2+5 )
2
−6 x +16 x+ 15
f ' ( x )= 2
( 2 x 2 +5 )
Chain Rule

Combination of general power rule

Example 1.
3
y= (3 x +2 )
3 −1 d
¿ 3 ( 3 x+2 ) ∙ (3 x+ 2)
dx
2
¿ 3 ( 3 x+2 ) ∙ 3(1)+ 0
2
¿ 3 ( 3 x+2 ) ∙ 3
2
y '=9 ( 3 x +2 )