Chained 1011818

Suppose g is differentiable at x and f is differentiable of kN.

Then fog is differentiable at x ant

tog) IN -- f 'll 9TH .

Intuitiveness : Write y .

- flew and u -

- geo . Then in Leibniz 's rotation.the chain role is :

Hittite .

It is tempting to just cancel the du 's.

but' '

II is just a symbol for something we do to a

function of u handy differentiation).

It is not a real fraction,

so cancelling it doesn't make sense .

The book gives a real proof in Section 3.4.

Erland find steer .

Solution refl is a composition of two functions.

'

If we let few be the"

inner function"

FINE x 't I

at flu be the " outer function"

flu =P,

Then footwear .

So wecan use the chain

ruler. We just need to know f ' and d

'. Then

we can plug everything inpower rule lower and sun rules

free fare # emit he- Hifi fruitful E2x

Therefore,

tyre - Hod'

N

Phe in our form was

= fibro . girl

for fig .

and o'FEET. 2x

=LNfl

.

temple.

find ¥ sink.

There are fatleast two ways we could do this : Product rule or chain role.

If we use the product rule,

he can compute :

¥ Sliestfine

. sink )

-

- thesheIsmet

sine #sine )

= cousin et sine cost

= 25in# se.

Let's check that the chain rule agrees with this answer.

We apply it with N and sink

¥ sink - Hsing '

differentiate the 728inNo# sink )outer severe

first

=

zgnecosk.

temple find

Lacosse:You could still use the product rule

, in a

"

chain"

¥Coste= ¥Kosercos * cos

= ¥cose) hose . cost ) those rose . rosel )

=- sine . cos 't those # cost dos et rosette cost )

= - costs int t cost f Sinko set rose f sine ))

=- cosy sink - sink COSH - sine cosy

.

-- 3 cosy sink

.

Watch how much easier the chain rule is :

¥ cosh -

- tf @set

power rule ?- 3 N'

cost )on therobe

=3 cost I - Sind

= - 3054 sing.

✓

Longview You can differentiate functions that look like

flysheet)

with the chain role,

or even longer chains.

for this one . you get i

Fetherolf forwent Gernot

-

- f'll Well . give . tired.

We used the chain rule twice.

Example find Ie sinksAmrein.

Solution

t sin , karate cos Leos

www.ftueosltmkl/--eoskosHnWlif-sinKanellgtetmk

= - rosko skin sinHarrel . see 't .

Combining.

'

Eiland find Tereus les - NVM

Solution ! first .use the product role

,then use the chain role on each piece :

Just product role

Hunslet # Mit#knishes- run

' teens return )

Now use the chaina snailrule on both derivatives=/ . 2) . It '

- et IftGetlls . 14h13 - xtll ?

(3×2-11)=1012tell 443 - til"

t 413k - I fuel Is IN - yup .

We can factor out a common

IN1/4/43 - HIP to simplify :

= 2. IS - tell t213eel Ike till Evil 'll '- HIP .=L

. Get the 't be ' -9k - 2) full MM - HIP

=L Has the . let 3) kettles - http .

Imagine doing this without the chain role !

team.

find Ta'

.

This time.

first do the chain role because the power is on the outside.

Then to the quotient role.

'

¥E¥i .

- if filteril-

- off I'

.

KittieGuy

'

=

h's eius si combine this suite the ,.

4.

see-218

=

get'

Exponentials We can now compute ¥6 " for any boo.

Remember

beefn

b

!Therefore . raising both sitesto the e gives

b' yetnot=p

Knox

Thus we can apply the chain role with e' and until :

¥6 's tf elm blue

- unble dbecause skeeterIup .

*Hnbit

= elhb InH

-

- b'

Inlol .

So.

for Hamp le

¥2 '-

- UH21.

Snitched ! This formula for bee should give Feet.

Indeed

,

¥ et-

- et . Chef et,

because he -1.

thplicitdifferentiation Sometimes we are given on expression involving Kandy which is more

complicated in y than

f- f CN.

For instance.

consider the equation for the circle of radius Si

uterus.

There is no way to mate this the graph ofa function because it fails the vertical line test

.

Yet

we can still make sense of II at any point .

To to this.

Apply ¥ to both sides and use the doin rule on any expression involving ti

Hurt feedI I

It' t

*Y

'-

- O

2kt 24 . ¥ - O

2e ¥ = - 2x.

if - e

This givesarelation between t

,t and ¥ . But what does It mean if y is not a function of l?

Well,

if I'm given any point call on my curve

.then if I ignore part of my curve away

from call, what's left my be the graph of a function with the point 4,61 on it .

For eeamplo.

if fall M) on my circle.

then I can ignore the bottom half of the circle where

13.91 does not lie :

i i

to:-.#it .

This "

top half"

is the graph of a function f because it passes the vertical line test.

In fort,

five Ffa.

and our relation I ¥ = - t is telling us that for my point law onthe top half of the circle I for

instance.

for 174 ) ) we have : b. FIN = - a I in case fable out this gives

4ft③

=-3or five } .

I