Section 3.4: Derivatives of Trigonometric Functions

[Pages:3]c Dr Oksana Shatalov, Spring 2012

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Section 3.4: Derivatives of Trigonometric Functions

It is important to remember that everything for six trigonometric functions (sin x, cos x, tan x, cot x, csc x, sec x) will be done in radians.

EXAMPLE 1. Compute:

THEOREM 2. Proof

(a) lim sin x =

x0

sin x lim = 1, x0 x

(b) lim cos x =

x0

cos x - 1

lim

= 0.

x0 x

EXAMPLE 3. Find these limits: sin(5x)

(a) lim x0 x

sin(9x) (b) lim

x0 sin(7x)

x (c) lim

x0 sin(4x)

Conclusion: If a, b = 0 then

sin(ax)

lim

=,

x0 x

x

lim

=,

x0 sin(ax)

sin(ax)

lim

=

x0 sin(bx)

c Dr Oksana Shatalov, Spring 2012

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1

(d)

lim

x0

x2

cot2(3x)

cos x - 1 (e) lim

x0 sin x

EXAMPLE 4. Find the following derivatives: d

(a) sinx = dx

Remark Similarly one can get (cos x) = - sin x. d (b) tanx = dx

Derivatives of Trig Functions (memorize these!)

d sinx =

dx

d cosx = - sin x

dx

d tanx =

dx

d cscx = - csc x cot x

d secx = sec x tan x

d cotx = - csc2 x

dx

dx

dx

c Dr Oksana Shatalov, Spring 2012

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EXAMPLE 5. Find the derivative of these functions.

(a) y = cot x + 5 sec x + x x

cos x (b) f (x) =

1 + sin x

EXAMPLE

6.

Find

the

equation

of

the

tangent

line

to

the

graph

of

function

y

= x2 sin x

at

x

=

. 4

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