Differentiation Rules

1
Differentiation Rules

• Basic Differentiation RulesDerived
  Differentiation RulesSpecial Function Rules

2
The Definition of the Derivative
Recall equivalent definitions
1
2
3
Basic Differentiation Rules
1
The derivative of the function f(x)x is 1.
2
3
The Product Rule
The Chain Rule
4
These are the basic differentiation rules which
imply all other differentiation rules for
rational algebraic expressions.
4
Derived Differentiation Rules
5
The Quotient Rule. Follows from the Product Rule.
Inverse Function Rule. Follows from the Chain
Rule.
6
5
Special Function Rules
12
7
8
13
9
14
10
15
11
6
Proof of the Product Rule (1)
Product Rule
First Proof
7
Proof of the Product Rule (2)
Product Rule
Second Proof
8
Proof of the Chain Rule
Chain Rule
Proof
9
Proof of the Quotient Rule
Quotient Rule
Proof
The Quotient Rule follows from this.
10
Proof of the Inverse Function Rule
Inverse Function Rule
Proof
The Inverse Function Rule follows from this.
Derivation differentiation rules for the inverse
trigonometric functions are good examples of the
usage of this formula. They will be presented
later.
11
Differentiation of Power Functions
Formula
Proof for positive integer values of r
Proof by Induction
1) r1
2) Induction Assumption True for rm
3) rm1
We will later see how the above general rule
follows from the differentiation rules for the
exponential function and its inverse function.
12
Differentiation of the Sine Function
Formula
Proof
13
Lemma for the Cosine function
Lemma
Proof
Using the trigonometric formula
we get
Hence
14
Differentiation of the Cosine Function
Formula
Proof
2) the Chain Rule, and
3) the Rule for differentiating the Sine
Function.
15
Differentiation of the Tangent Function
Formula
Proof
16
Inverse Trigonometric Functions (1)
Formula
Proof
17
Inverse Trigonometric Functions (2)
Formula
Proof
18
Differentiation of Exponential Functions
Formula
Proof
19
General Exponential Functions
Formula
Proof
20
Differentiation of the Logarithm
Formula
Proof
21
Differentiation of General Powers
Formula
Proof