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Rules for Derivatives: Finding the Derivative of Functions, Study notes of Calculus

A comprehensive tutorial on the rules for derivatives, including the power rule, sum rule, constant coefficient rule, chain rule, u-sub, product rule, quotient rule, and special rules. It covers various examples and explanations to help understand the concepts.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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Rules for Derivatives
James S__ Jun 2010 r6
Derivatives (Dx):
In this tutorial we will use Dx for the derivative (𝒅𝒅
𝒅𝒅𝒅𝒅). Dx indicates that we are taking the
derivative with respect to x. 𝑓𝑓(𝑥𝑥) is another symbol for representing a derivative.
The derivative represents the slope of the function at some x, and slope represents a rate of
change at that point.
The derivative (Dx) of a constant (c) is zero.
Ie: y = 3 since y is the same for any x, the slope is zero (horizontal line)
Power Rule: The fundamental tool for finding the Dx of f(x)
multiply the exponent times the coefficient of x and then reduce the exponent by 1
Ex: 𝑫𝑫𝒅𝒅 [𝒅𝒅𝟑𝟑] 𝑓𝑓(𝑥𝑥)= 3𝑥𝑥2 𝑑𝑑𝑥𝑥 = 3𝑥𝑥2
𝑫𝑫𝒅𝒅 [𝒅𝒅𝟑𝟑+𝟓𝟓] 𝑓𝑓(𝑥𝑥)= 3𝑥𝑥2 𝑑𝑑𝑥𝑥 = 3𝑥𝑥2
* [dx represents the derivative of what is inside (x), which is usually 1 for simple functions, the dx
must always be considered and is always there, even if it is only 1]
Sum Rule: The Dx of a sum is equal to the sum of the Dx’s
Ex: 𝐷𝐷𝑥𝑥 [3𝑥𝑥2+ 2𝑥𝑥+ 3] 𝑓𝑓(3𝑥𝑥2)+𝑓𝑓(2𝑥𝑥)+ 𝐹𝐹(3)= 6𝑥𝑥+ 2 + 0
Constant Coefficient Rule: The Dx of a variable with a constant coefficient is equal to the
constant times the Dx. The constant can be initially removed from the derivation.
Ex: 𝐷𝐷𝑥𝑥[ln(4)𝑥𝑥2]=ln(4)𝐷𝐷𝑥𝑥[𝑥𝑥2]=ln(4)2𝑥𝑥= 2 ln(4)𝑥𝑥=ln(4)2𝑥𝑥=ln(16)𝑥𝑥
Chain Rule: There is nothing new here other than the dx is now something other than 1. The dx
represents the Dx of the inside function g(x). It is called a chain rule because you have to consider the
dx as not being 1 and take the Dx of the inside also.
Ex: Dx (sin(3x)) = cos(3x) dx* = 3 cos(3x) * [dx is g’(3x) = 3]
* [the dx here is g’(x)]
Ex: Dx [(3x2+2)2] = 2(3x2+2) dx* = 2( 3x2+2 ) (6x) = (6x2 + 4)(6x) = 36x3
+ 24x
*[dx is Dx (3x2
+ 2) = 6x] notice we used the Power Rule along with the Chain Rule
𝐷𝐷𝑥𝑥 [𝑥𝑥𝑛𝑛]𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑑𝑑𝑦𝑦
𝑓𝑓(𝑥𝑥)= 𝑛𝑛𝑥𝑥𝑛𝑛1
𝐷𝐷𝑥𝑥[𝑓𝑓(𝑥𝑥)+ 𝑔𝑔(𝑥𝑥)] 𝑓𝑓(𝑥𝑥)+ 𝑔𝑔(𝑥𝑥)
𝐷𝐷𝑥𝑥[3𝑥𝑥2]= 3(𝐷𝐷𝑥𝑥[𝑥𝑥2])
𝐷𝐷𝑥𝑥 [𝑓𝑓�𝑔𝑔(𝑥𝑥)�⟹𝑓𝑓𝑔𝑔(𝑥𝑥)�𝑑𝑑𝑥𝑥 =𝑓𝑓𝑔𝑔(𝑥𝑥)�𝑔𝑔(𝑥𝑥)
pf2

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Rules for Derivatives

James S__ Jun 2010 r

Derivatives (Dx) :

  • In this tutorial we will use Dx for the derivative (

𝒅𝒅

𝒅𝒅𝒅𝒅

). Dx indicates that we are taking the

derivative with respect to x. 𝑓𝑓

(𝑥𝑥) is another symbol for representing a derivative.

  • The derivative represents the slope of the function at some x, and slope represents a rate of

change at that point.

  • The derivative (Dx) of a constant (c) is zero.

 Ie: y = 3 since y is the same for any x, the slope is zero (horizontal line)

Power Rule: The fundamental tool for finding the Dx of f (x)

  • multiply the exponent times the coefficient of x and then reduce the exponent by 1

Ex: 𝑫𝑫𝒅𝒅 [𝒅𝒅

𝟑𝟑

] → 𝑓𝑓

2

2

𝑫𝑫𝒅𝒅 [𝒅𝒅

𝟑𝟑

+ 𝟓𝟓] → 𝑓𝑓

2

2

  • [dx represents the derivative of what is inside (x), which is usually 1 for simple functions, the dx

must always be considered and is always there, even if it is only 1]

Sum Rule: The Dx of a sum is equal to the sum of the Dx’s

Ex: 𝐷𝐷𝑥𝑥

[

2

+ 2𝑥𝑥 + 3]

2

Constant Coefficient Rule: The Dx of a variable with a constant coefficient is equal to the

constant times the Dx. The constant can be initially removed from the derivation.

Ex: 𝐷𝐷𝑥𝑥

[ln(4) 𝑥𝑥

2

] = ln(4) 𝐷𝐷𝑥𝑥

[

2

] = ln(4) ∗ 2 𝑥𝑥 = 2 ln(4) 𝑥𝑥 = ln(4)

2

𝑥𝑥 = ln(16) 𝑥𝑥

Chain Rule: There is nothing new here other than the dx is now something other than 1. The dx

represents the Dx of the inside function g (x). It is called a chain rule because you have to consider the

dx as not being 1 and take the Dx of the inside also.

Ex: Dx ( sin (3x)) = cos( 3x ) dx* = 3 cos(3x) * [dx is g’( 3x ) = 3]

  • [the dx here is g’(x) ]

Ex: Dx [(3x

2

2

] = 2(3x

2

+2) dx* = 2( 3x

2

+2 ) (6x) = (6x

2

  • 4)(6x) = 36x

3

  • 24x

*[dx is Dx (3x

2

    1. = 6x] notice we used the Power Rule along with the Chain Rule

𝐷𝐷𝑥𝑥 [𝑥𝑥

𝑛𝑛

]

𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑑𝑑𝑦𝑦

𝑛𝑛 − 1

[

)]

𝐷𝐷𝑥𝑥[ 3 𝑥𝑥

2

] = 3 (𝐷𝐷𝑥𝑥[𝑥𝑥

2

])

𝐷𝐷𝑥𝑥 [𝑓𝑓�𝑔𝑔(𝑥𝑥)� ⟹ 𝑓𝑓

Rules for Derivatives

James S__ Jun 2010 r

U-sub: This is when you let some letter equal the whole inside quantity. It can be very useful in a

Chain Rule situation.

Ex: Dx [(sin(x))

3

Now we have: Dx [U

] ►If we let U = sin(x) ⟹ then du = cos(x)

3

] = 3U

2

⟹ 3[sin(x)]

du

2

[cos(x)] ►substitute back in for U and du

Product Rule: The Dx of a product is equal to the sum of the products Dx of each factor

times the other factor.

Ex: 3 𝑥𝑥

2

𝑥𝑥

𝑥𝑥

2

𝑥𝑥

Quotient Rule : Dx (numerator) times the denominator minus Dx (denominator) times the

numerator, divided by the denominator squared. This is a variation of the Product Rule.

Ex: 𝐷𝐷𝑥𝑥 �

sin (𝑥𝑥)

3 𝑥𝑥

cos (𝑥𝑥

)( 𝑥𝑥

) −sin (3𝑥𝑥

)(3)

(3𝑥𝑥)

2

3 𝑥𝑥𝑥𝑥𝑥𝑥𝑦𝑦 (𝑥𝑥)−3sin (3𝑥𝑥)

9 𝑥𝑥

2

𝑥𝑥𝑥𝑥𝑥𝑥𝑦𝑦 (𝑥𝑥)−sin (3𝑥𝑥)

3 𝑥𝑥

2

Special Rules:

[ln(

)] =

1

𝑥𝑥 𝑦𝑦𝑛𝑛 (𝑦𝑦)

1

𝑥𝑥

Ex: 𝐷𝐷𝑥𝑥

[ln(sin( 𝑥𝑥

))] =

1

sin (𝑥𝑥)𝑦𝑦𝑛𝑛 (𝑦𝑦)

cos(𝑥𝑥

cos (𝑥𝑥)

sin (𝑥𝑥)

= cot(𝑥𝑥)

Ex: 𝐷𝐷𝑥𝑥 [log(3𝑥𝑥

2

)] =

1

(3𝑥𝑥

2

)(𝑦𝑦𝑛𝑛 10)

3

(𝑥𝑥)(𝑦𝑦𝑛𝑛 10)

• 𝐷𝐷𝑥𝑥 [𝑦𝑦

𝑥𝑥

] = 𝑦𝑦

𝑥𝑥

𝑑𝑑𝑥𝑥 ln(𝑦𝑦)

Ex: Dx [3e

4x

] = 3[(e

4x

)ln(e)] = 12(e

4x

Ex: 𝐷𝐷𝑥𝑥 [

𝑥𝑥

2

+5𝑥𝑥

] = (

𝑥𝑥

2

+5𝑥𝑥

)[ln(13)](2𝑥𝑥 + 5)

𝐷𝐷𝑥𝑥 [𝑓𝑓(𝑥𝑥) • 𝑔𝑔(𝑥𝑥)] ⟹ [𝑓𝑓

(𝑥𝑥)𝑑𝑑𝑥𝑥]

[ 𝑔𝑔(𝑥𝑥)]

2

[

𝑏𝑏

𝑏𝑏

]

[

𝑥𝑥

]

𝑥𝑥

(ln