1

Derivatives of Trigonometric Functions

Q. Where did the formulas for the derivatives of sinx and cosx come from?

A1. Graphically:

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

Lecture 22

2

xxxdx

dx

dx

d

xh

xhxx

dx

dh

sincossincos

cos......sinsin

limsin

122

0

A2. By the Definition of the Derivative:

Q. Where did the formulas for the derivatives of sinx and cosx come from?

3

Derivative of the Sine

sin cos

sin cos

dx x

dxd du

u udx dx

Generalized Rule

Ex. 3sin 5 2d

x xdx

3 2cos 5 2 15 2x x x

4

Derivative of the Cosine

cos sin

cos sin

dx x

dxd du

u udx dx

Generalized Rule

Ex. cos 2xde x

dx

sin 2 2x xe x e

5

Derivative of the Tangent

2

2

tan sec

tan sec

dx x

dxd du

u udx dx

Generalized Rule

Ex. tan lnd

xdx

2 1sec ln x

x

6

Derivative of the Secant

sec sec tan

sec sec tan

dx x x

dxd du

u u udx dx

Generalized Rule

Ex. sec sind

xdx

sec sin tan sin cosx x x

7

Derivative of the Cosecant

csc csc cot

csc csc cot

dx x x

dxd du

u u udx dx

Generalized Rule

Ex. 2csc 1d

xdx

2 2csc 1 cot 1 2x x x

8

Derivative of the Cotangent

2

2

cot csc

cot csc

dx x

dxd du

u udx dx

Generalized Rule

Ex. cot 5xd

dx

2csc 5 5 ln 5x x

9

Derivatives

Ex. csc 4 1 sin 3 1d

x xdx

4csc 4 1 cot 4 1 sin 3 1

3cos 3 1 csc 4 1

x x x

x x

10

Derivatives

Ex. sec ; find dy

x ydx

1 sec tandy

y ydx

1cos cot

sec tan

dyy y

dx y y

Implicit differentiation

11

Problem 53 on page 562(similar to Section 9.2 Problem. 54)

53. Cost The cost in dollars of Dig-In brand shovels is given by where t is time in years since January 1, 2002. How fast, in dollars per week, is the cost increasing each October 1?

750253 .sin. ttc

12

Solution:

Given: 750253 .sin. ttc

Find: How fast, in dollars per week, is the cost increasing each Oct. 1? That is,

$/week$/yr 750 ??. c

wkyr

c

/.$/.$.

cos.

..cos..

4209921107

007

750750207750 Thus,

750207

2750253

.cos.

.cos.

t

ttc