3.5 DERIVATIVES OF TRIG FUNCTIONS

Some needed trig identities:1cossin 22 xx

xx 22 tan1sec

xx 22 cot1csc

xxx cossin22sin

xxx 22 sincos2cos

yxyxyx sincoscossin)sin( yxyxyx sinsincoscos)cos(

Trig DerivativesGraph y1 = sin x and y2 = nderiv (sin x)

What do you notice?

xxdx

dcos)(sin

Proof Algebraically

h

xhxx

dx

dh

sin)sin(lim)(sin

0

(use trig identity for sin(x + h))

h

xxxh

sinsinh)coscosh(sinlim

0

h

xxxh

sinhcossincoshsinlim

0

h

xxh

sinhcos)1(coshsinlim

0

Proof Algebraically

h

x

h

xh

sinhcos)1(coshsinlim

0

hx

hx

hh

sinhcoslim

1coshsinlim

00

0 1

xcos

Trig DerivativesGraph y1 = cos x and y2 = nderiv (cos x)

What do you notice?

xdx

dsin(cos)

Proof Algebraically

h

xhxx

dx

dh

cos)cos(lim)(cos

0

(use trig identity for cos(x + h))

h

xxxh

cossinh)sincosh(coslim

0

h

xxxh

sinhsincoscoshcoslim

0

h

xxh

sinhsin)1(coshcoslim

0

Proof Algebraically

h

x

h

xh

sinhsin)1(coshcoslim

0

hx

hx

hh

sinhsinlim

1coshcoslim

00

0 1

xsin

Other Trig Derivatives

x

x

dx

dx

dx

d

cos

sin)(tan (quotient

rule)

2)(cos

)sin(sin)(coscos

x

xxxx

x

xx2

22

cos

sincos (trig id cos2x + sin2x

= 1)

x2cos

1 x2sec

Other Trig Derivatives

x

x

dx

dx

dx

d

sin

cos)(cot (quotient

rule)

2)(sin

)(coscos)sin(sin

x

xxxx

x

xx2

22

sin

cossin

x

xx2

22

sin

)cos(sin1 x2csc

x2sin

1

Other Trig Derivatives

xdx

dx

dx

d

cos

1)(sec (quotient

rule)

2)(cos

)sin(1)0(cos

x

xx

x

x2cos

sin0

xx

x

cos

1

cos

sin xx sectan

Other Trig Derivatives

xdx

dx

dx

d

sin

1)(csc (quotient

rule)

2)(sin

)(cos1)0(sin

x

xx

x

x2sin

cos0

xx

x

sin

1

sin

cos

xx csccot

ExampleFind an equation of the tangent line to the function

f(x) = sec x at the point

2,3

derivative slope

xxxfm tansec)('

3tan

3sec

3'

f

32 32 (slope)

bmxy b

3322

b3

322

b

3

326

3

32632

xy

ExampleFind the second derivative of y = csc x.

xxy cotcsc'

)cot(csccot)csc(csc'' 2 xxxxxy (Product rule)

xxxy 23 cotcsccsc''