LECTURE: 3-3 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS

Recall last time we found ddx (sinx) = cosx and d

dx (cosx).

Example 1: Using the derivative of sinx and cosx find derivatives of:

(a) y = cotx (b) y = cscx

Derivatives of Trigonometric Functions:

• ddx (sinx) =

• ddx (cosx) =

• ddx (tanx) =

• ddx (cscx) =

• ddx (secx) =

• ddx (cotx) =

Example 2: Find the second derivatives of the following functions:

(a) g(t) = 4 sec t+ tan t. (b) y = x2 sinx.

Example 3: Find an equation of the tangent line to the curve y =1

sinx+ cosxat the point (0, 1).

UAF Calculus I 1 3-3 Derivatives of Trigonometric Functinos

= COSI =I

six six

y'

I sinxccosx )'

- co , xc six )'

y)

=six ¥-1

( sin

Tsin Tsin

= - sin2x-cos= -s.IS#x=-eotxcsaxJSin2x

= =-c

cos X - cotxcscx- sinx secxtonx

seek - csix

first•

547=4sect Emttsec 't y

'= 2xsinx + x7osx

=sect(4tuttsec =x(2sin*txcos

yr =Cinxtcosx )C0 ) - I ( cosy - six )

-

-( sinxtcosx )2

=

sinx-cosx-x-oi.jcd-syjfo.IO?-o,s=-Is=-I(sinxtcosxP

y - I = - I ( x - O )

y-

Example 4: For what values of x does the graph of f(x) = x+ 2 sinx have a horizontal tangent?

Example 5: Differentiate f(x) =secx

1� tanxand determine where the tangent line is horizontal.

Generalized Product Rule: How does the product rule genearlize to more than two functions? For example, whatis the derivative of y = f(x)g(x)h(x)?

Example 6: Differentiate h(✓) = ✓2 tan ✓ sec ✓.

UAF Calculus I 2 3-3 Derivatives of Trigonometric Functinos

f'

C D= I +2 cos x = O

con .⇒ . "¥x:÷÷÷ ⇒ .

f'

C ⇒ = ( I - tax ) Csecxtonx ) - secx ( - Seis )-

Cl - tan x )-

cats ? I

=secx ( tax - tank t seok ) I + tar ? see

'

-

( I - tax )2

= Secx ( tax + D see -

- O X - Tues- ten x -11=0

( I - tan xpfan ×=

- I

¥374x=3tf C x ) . ( s hlx ))

powerrule within power rule !

D= f G) ( gcx ) .li Ix ) -15634×7 ) -1 f' Cx ) ( glx )hLx7 )

( )

h' ④ = of ( tmo seco

't

20-tmo-se-O-fftano.seco-tuo-tsefo-seco-3-2.frtonoseco

÷:÷::::÷÷:÷:

Example 7: Find the 51st derivative of f(x) = sinx. Specifically, find the first four or five derivatives and look fora pattern.

Example 8: A mass on a spring vibrates horizontally on a smooth level surface. Its equation of motion is x(t) =8 sin t, where t is in seconds and x is in centimeters.

(a) Find the velocity at time t.

(b) Find the position and velocity of the mass at time t = 2⇡/3. In what direction is it moving at this time?

Example 9: A ladder 12 feet long rests against a vertical wall. Let ✓ be the angle between the top of the ladder andthe wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides awayfrom the wall, how fast does x change with respect to ✓ when ✓ = ⇡

6 .

UAF Calculus I 3 3-3 Derivatives of Trigonometric Functinos

f'

L x ) -_ cos Cx )

f-"

( × ) = - sin X hmm . 51 is odd .Either

Cos x or - cos x

f" '

( x ) = - cost52 is divisible by

,so is

fu ) ( ⇒ = sire

y

so fC5' KD =f' "

C x ) = -c

positive x ,to the right

x' G) = 8 cost

--

X (2 "G ) x

'( 273 )

position : x ( F) = 8 sin (2*5)=8 Fsz =453cLvelocity : x'

(F) = 8 cos = 8C -

I

s ) = -4cnLeft

- - de

sin ⑤ = ¥do

- ⇒ x = 12 sinox

diff -- If C 12 sin A

= 12 cos ⑦

x' ( %) = 12 cos €1213=6r3