Summary Rules and Techniques for Differentiation

Rules and Techniques for Differentiation
Anibal Garcia
August 7 2020



Contents

I Differentiation 4
1 Simple Rules of Differentiation 4

2 The Power Rule 5
2.1 The Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Proof of The Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Linearity of the Derivative 7

4 The Sum Rule 8
4.1 The Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Proof of the Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 The Product Rule 9
5.1 The Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2 Proof of the Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.3 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 The Quotient Rule 12
6.1 The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Proof of the Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 The Chain Rule 14
7.1 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.2 Proof of the Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.3 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Derivatives of Exponential Functions 21
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
8.2 A function with its own derivative, f (x) = ex . . . . . . . . . . . . . . . . 22
8.3 The derivative of ef (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1

, 8.3.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.3.2 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8.4 The derivative of ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.5 The derivative of af (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.5.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.5.2 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Derivatives of Logarithmic Functions 29
9.1 The derivative of ln x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2 The derivative of ln f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2.2 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.3 The Derivative of loga x . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.3.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.4 The Derivative of loga f (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.4.2 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10 Derivatives of Trigonometric Functions 35
10.1 The Derivative of sin x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10.2 The Derivative of cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10.3 The Derivative of tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10.3.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10.4 The Derivative of sin[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10.5 The Derivative of cos[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10.6 The Derivative of tan[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10.7 Solved Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
10.8 The Derivative of csc x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
10.8.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
10.9 The Derivative of sec x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
10.9.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
10.10The Derivative of cot x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
10.10.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
10.11Derivative of csc[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
10.12Derivative of sec[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
10.13Derivative of cot[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11 Derivatives of Inverse Trigonometric Functions 42
11.1 Derivative of arcsin x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.2 Derivative of arccos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.3 Derivative of arctan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
11.4 Derivative of arcsin[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
11.4.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
11.5 Derivative of arccos[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
11.6 Derivative of arctan[f (x)] . . . . . . . . . . . . . . . . . . . . . . . . . . 44


2

,12 Second and Higher Derivatives 46




3

,Part I
Differentiation
1 Simple Rules of Differentiation

Differentiation is the process of finding a derivative or gradient function.
Given a function f (x) we obtain f 0 (x) by differentiating with respect to the variable x.

There are a number of rules associated with differentiation. These rules can be used to
differentiate more complicated functions without having to use first principles.

We can summarize the following rules:

f (x) f 0 (x) Name of Rule
c (a constant) 0 Differentiating a Constant
xn nxn−1 Differentianting xn / Power Rule
cu(x) cu0 (x) Constant times a function
u(x) + v(x) u0 (x) + v 0 (x) Addition Rule


The last two rules can be proved with the first principles of differentiation:




4

,2 The Power Rule
2.1 The Power Rule
We start with the derivative of a power function, f (x) = xn . Here n is a number of
any kind: integer, rational, positive, negative, even irrational, as in xπ . The formula to
differentiate xn :
d n
x = nxn−1
dx



2.2 Proof of The Power Rule
We can prove this with the definition of the derivative and the binomial expansion.
Find the derivative of f (x) = xn :

First, we know the definition of the derivative is:
f (x + h) − f (x)
lim
h→0 h
We apply this to xn :
(x + h)n − xn
lim
h→0 h
We know that the binomial expansion for (x + h)n is:
n  
n
X n n−r r
(x + h) = x h
r=0
r
n          
X n n−r r n n n n−1 n n−2 2 n n
x h = x + x h+ x h + ... + h
r=0
r 0 1 2 n
We introduce this expansion into the limit:
n
x + n1 xn−1 h + n2 xn−2 h2 + ... + n

n


n
0 n
h − xn
lim
h→0 h
We cancel out terms:
n  n n n
n


n−1
n−2 2
n 
n
−x
lim 0x + 1
x h+ 2
x h + ... + n
h
h→0 h
n n n




1
xn−1 h + 2
xn−2 h2 + ... + n
hn
lim
h→0 h
We then cancel h:
h hn−1
n n n

n−1

2
n−2 

>
n
1
x h+
 2
x h + ... + n
h

lim
h→0 h

     
n n−1 n n−2 n n−1
lim x + x h + ... + h
h→0 1 2 n

5

, We could then factorize:
      !
n n−1 n n−2 n n−1
lim x +h x + ... + 1
h→0 1 2 n

We can then evaluate the limit:

     :!0
   
n n−1 n n−2  n n−1
 n n−1
lim x +h x +
... + 1 = x
h→0 1 2  n 1
 


∴ f 0 (x) = nxn−1

2.3 Solved Exercises
1. Find the derivatives of the given functions.

(a) x100 (d) xπ
3
(b) x−100 (e) x 4
1 √
(c) x5
(f) 3 x

(a)
d 100
x = 100x99
dx
(b)
d −100
x = −100x−101
dx
(c)
d 1 d −5
≡ x = −5x−6
dx x5 dx
(d)
d π
x = πxπ−1
dx
(e)
d 3 3 3 3 5
x 4 = x( 4 −1) = x− 4
dx 4 4
(f)
d √ d 1 3 1
3 x≡ 3x 2 = x− 2
dx dx 2




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