2

0

2

Consider the function siny

We could make a graph of the slope: slope

1

0

1

0

1Now we connect the dots!The resulting curve is a cosine curve.

sin cosd

x xdx

2.3 Derivatives of Trigonometric Functions

2.3 Derivatives of Trigonometric Functions

h

xhxx

dx

dh

sin)sin(limsin

0

h

xxhhxx

dx

dh

sincossincossinlimsin

0

h

xh

h

hxx

dx

dhh

cossinlim

)1(cossinlimsin

00

h

xhhxx

dx

dh

cossin)1(cossinlimsin

0

Proof

2.3 Derivatives of Trigonometric Functions

h

xh

h

hxx

dx

dhh

cossinlim

)1(cossinlimsin

00

= 0 = 1

sin cosd

x xdx

2.3 Derivatives of Trigonometric Functions

h

xhxx

dx

dh

cos)cos(limcos

0

h

xxhhxx

dx

dh

cossinsincoscoslimcos

0

h

xh

h

hxx

dx

dhh

sinsinlim

)1(coscoslimcos

00

h

xhhxx

dx

dh

sinsin)1(coscoslimcos

0

Find the derivative of cos x

2.3 Derivatives of Trigonometric Functions

= 0 = 1

h

xh

h

hxx

dx

dhh

sinsinlim

)1(coscoslimcos

00

cos sind

x xdx

We can find the derivative of tangent x by using the quotient rule.

tand

xdx

sin

cos

d x

dx x

2

cos cos sin sin

cos

x x x x

x

2 2

2

cos sin

cos

x x

x

2

1

cos x

2sec x

2tan secd

x xdx

2.3 Derivatives of Trigonometric Functions

=

=

=

=

=

Derivatives of the remaining trig functions can be determined the same way.

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx

2.3 Derivatives of Trigonometric Functions

2.3 Derivatives of Trigonometric Functions

dy dy du

dx du dx Chain Rule:

2.4 Chain Rule

Let h(x) = f(g(x)) (also known as )

Then h’(x) = (f(g(x))’ =

f g

at at xu g xf g f g

)('))((' xgxgf

Here is a faster way to find the derivative:

2sin 4y x

2 2cos 4 4d

y x xdx

2cos 4 2y x x

Differentiate the outside function...

…then the inside function

2.4 Chain Rule

‘

‘

2cos 3d

xdx

2cos 3

dx

dx

2 cos 3 cos 3d

x xdx

2cos 3 sin 3 3d

x x xdx

2cos 3 sin 3 3x x

6cos 3 sin 3x x

The chain rule can be used more than once.

(That’s what makes the “chain” in the “chain rule”!)

2.4 Chain Rule

=

=

=

=

=

Derivative formulas include the chain rule!

1n nd duu nu

dx dx sin cos

d duu u

dx dx

cos sind du

u udx dx

2tan secd du

u udx dx

etcetera…

2.4 Chain Rule

2.4 Chain Rule

Find

)3cos( 2 xxy )16)(3sin( 2 xxxdx

dy

))sin(cos(xy

)24(cos 33 xxy

)sin)(cos(cos xxdx

dy

)212))(24sin()(24(cos3 2332 xxxxxdx

dy

))24sin()(24(cos)636( 3322 xxxxxdx

dy

dx

dy