3.5 Differentiation Formulas for Trig Functions: Sine and ... - UToledo

[Pages:2]3.5 Differentiation Formulas for Trig Functions: Sine and Cosine: Recall

from Section 2.4 that

lim sin h = 1 0 h

(by the Squeeze Theorem) and that we derived

lim cos h - 1

h0

h

=

lim

h0

(cos

h - 1))(cos h h(cos h + 1)

+

1)

=

lim

h0

(cos h)2 - 1 h(cos h + 1)

=

lim

h0

-(sin h)2 h(cos h + 1)

=

lim

h0

- sin h h

sin

h

cos

1 h

+

1

=

-1(0)(1/2)

=

0

Recall also the sum of the angles formula for sin x and cos x:

sin(A + B) = sin A cos B + sin B cos A cos(A + B) = cos A cos B - sin A sin B

Compute the difference quotient

sin(x + h) - sin x h

=

sin x cos h + sin h cos x - sin x h

=

sin

x(cos h

h

-

1)

+

sin

h cos h

x

=

sin

x cos

h- h

1

+

cos

x

sin h

h

Now let h approach 0.

d sin x = lim sin(x + h) - sin x = sin x lim cos h - 1 + cos x lim sin h = cos x

dx

h0

h

h0

h

h0 h

so that

d sin x = cos x dx

Similarly we can compute the difference quotient.

cos(x + h) - cos x = cos x cos h - sin x sin h - cos x

h

h

=

cos

x(cos h

h

-

1)

-

sin

h sin h

x

=

cos

x cos

h- h

1

-

sin

x

sin h

h

Now let h approach 0.

d dx

cos x

=

lim

h0

cos(x

+

h) h

-

cos x

=

cos x lim

h0

cos h - h

1

-

sin x lim

h0

sin h h

=

- sin x

so that

d dx

cos

x

=

-

sin

x

2

Example: Find the derivative of tan x.

Solution:

Recall

that

tan x

=

sin x cos x

so

that

by

the

quotient

rule

d dx

tan x

=

cos

x

d dx

sin

x

-

sin

x(

d dx

(cos x)2

cos x)

=

cos x cos x - sin x(- sin x) (cos x)2

=

1 (cos x)2

=

(sec x)2

Example: Solution:

Find the derivative Recall that cot x =

csooifnscxxotsxo.that

by

the

quotient

rule

d dx

cot x

=

sin

x

d dx

cos

x

-

cos

x(

d dx

(sin x)2

sin x)

=

sin x(- sin x) - cos x(cos x) (sin x)2

=

-1 (sin x)2

=

-(csc x)2

Example: Find the derivative of sec x = 1/ cos x Solution: By the reciprocal rule

d dx

sec x

=

-

d dx

cos

x

(cos x)2

=

sin x (cos x)2

=

sec x tan x

Example: Find the derivative of csc x = 1/ sin x Solution: By the reciprocal rule

d dx

csc x

=

-

d dx

sin

x

(sin x)2

=

- cos x (sin x)2

=

- csc x cot x

Function sin x cos x tan x cot x sec x csc x

Derivative cos x - sin x

(sec x)2 -(csc x)2 sec x tan x - csc x cot x

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