Lecture 9: Derivatives

Higher Derivatives !

, then successive derivatives can also be denoted byIf

Other common notations are

These are called, in succession, the first derivative, the second derivative, the third derivative, and so forth. The number of times that f is differentiated is called the order of the derivative. A general nth order derivative can be denoted by

Example: If then

Derivative of a Product !

Proof. 

Example Find if

Method 1. (Using the Product Rule)

Method 2. (Multiplying First)

Derivative of a Quotient !

Proof

for Example 3 Find

Example Find for

Solution.  Applying the quotient rule yields

y(x) =x4 � 3x2 + x + 2

x� 3

dy

dx=

d

dx

x4 � 3x2 + x + 2x� 3

=(x� 3) d

dx (x4 � 3x2 + x + 2)� (x4 � 3x2 + x + 2) ddx (x� 3)

(x� 3)2

=(x� 3)(4x3 � 6x + 1)� (x4 � 3x2 + x + 2) · 1

(x� 3)2

=4x4 � 6x2 + x� 12x3 + 18x� 3� x4 + 3x2 � x� 2

x2 � 6x + 9

=3x4 � 12x3 � 3x2 + 18x� 5

x2 � 6x + 9

Trigonometry

Definitions:

Trigonometric Identities !

A.2 Theorem. (Law of Cosines) If the sides of a triangle have lengths a, b, and c, and if is the angle between the sides with lengths a and b, then

double-angle formulas

product-to-sum formulas

Some useful Limits

we have already computed using squeezing theorem:

Proof:

3.5   Derivatives of Trigonometric Functions

Example:

Find if

Example:

FindExample

if

Solution