Idea ! we now have some nice rules for

calculating derivatives involving natural operations

( sums,differences

, products , quotients ) .

What about function composition ?

If Fcx )= fcgcx )) , how is F'Cx )

related to L'Cx ) and g'Cx ) ?

The chain rule : ifg

is differentiable at x and

f is differentiable at gcxl , then F-Cx ) -- fcgcxl ) is

differentiable at x and F'

is given by the product

F'Cx ) = L'Cgcx ) ) g'Cx )

In Leibniz notation,

if y-

- flu) and u=gCx ) , then

fit,= # EI

Proof :

fcgcxthll-hfcgcx.fcgcxthll-fcg.ge#xlgCxthl-gCx) h

Problem : what if gcxth) - gcx) -

- O ?

Consider two cases :g'

Cx ) # O and g'Cx) -_ 0

If g'Cx) # O, then gcxth) -gcx ) # O for any sufficiently

small h.

Moreover him gcxth) -gCxl=O sinceg is

differentiable and hence continuous

Then hi.am#m-hesH-=nl.i.mlggfIn;h!jffYhi..mo EYE

Then him. www.hesa#=nIimlggfInY!jffYti..moE''hee

= lim""Iim g¥)

d-70 d hao h

tD= glxthl -glx )

= f'Cgcxl ) g'Cx )

If g'Cx ) -- O ,

then we consider the possible cases :

if gcxth) -glx)-

- O,

then gcxthl -- g Cx) and fcgcxth) ) -- fcgcx ))

and fcgcxth ) ) - fcgcx )) =O and fcgcxthll-hfcgcx.cootherwise

,

we can use the same idea as

The chain rule : ifg

is differentiable at x and

f is differentiable at gcxl , then F-Cx ) -- fcgcxl ) is

differentiable at x and F'

is given by the product

F'Cx ) = L'Cgcx ))g'Cx ) or clog ) 'Cxl=fGCxHg'Cx)

In Leibniz notation,

if y-

- flu) and u=gCx ) , then

fit,= # EI

Example : Fcxl = Txt.

Find F'Cx ).

Ans :

Example : Fcxl = Txt.

Find F'Cx ).

Ans : Fcx ) -- foglx) , where fcx ) : Tx and gcxt -- x'et .

(Tx)-

- 2¥ g'

Cx) -

- 2x

Then F'Cx ) ' L'Cgcxllgtx ) = z¥+,

2x = ×¥

Example : differentiate y= coscx') and

y = cost ) ( = (coscx) )' )

.

Ans :

Example : differentiate y= coscx') and

y = costs ) ( = (coscx) )' )

.

Ans : fcx )-

- coscx ) gcx ) = x'

f.'

G) = - sink)g.'Cx) -- 2x

first case is fcgcxl)'

= ('

Cglx))gtx ) = - sink' ) - 2x

second case is gcflx))'

= g'Cfcx)) L'Cx) = Zcoscx) - C- sink ) )

Idea : if u is a differentiable function of x,

theny = sincu ) has derivative

d¥ = ddt d¥ = coscu) IF

Ify-

- costal,

then

d¥ = ddt d¥ = - sin cu) II

Last slide : u=x2

Ify

= u

"

and u=gCx ) ,

then

dat = d£ IF = nu"- '

g'Cx )

or day (un ) -- nun"

date

or daylight"

] = nglx)" - '

g'Cx )

Example : differentiate y= ( 2×4- x - l )

""

Ans :

Example : differentiate y = ( 2×4 - x - l )""

Ans : gcx ) -- 2x"- x - I

f- Egan"" ] = 2021gal

""

g'

Cx)

= 2021 ( 2×4 - x - I )""

( 8×3 - I )

Example : differentiate fcxl :#X - txt 3Ans :

Example : differentiate fcxl :#X - t Xt 3Ans : fat . ( x' txt

35¥

if glxl -- x'txt 's

(Cx)-

- ga-I

f 'm-

- I gcxi "gy×I= -1264×+3,312

( 2x -11 )

Example : differentiate gct ) -- ( )'

Ans :

Example : differentiate gct ) -- ( )'

Ans : get -- fats,

where fat : IIIT

(31-71161 - ht -hot)('

CH --

-( St-

H )

g 'H75eH5' yet) - 5 )"

( 3-1 't 1)2

Example : differentiate fcx)-

- (3×-1714×4+1)'

Ans :

Example : differentiate fct ) -- (31--17544+1)'

Ans : Lct ) : gH5hHP ,

where GH ) -- 3-1-17htt -- Htt

(G) = get)'

( 3h45 htt) ) t 2gHg'Ct) htt'

= (31-+7)<(344+1)-(41-3)) t 2 (3ft7)(3) (Htt )'

Ify

= e"

and u=gCx ) ,

then

U

daff = d£ IF = e g'

Cx )

or £×Ce4= e"

¥

or ⇐Ees '" ] = e'"'

g'Cx )

Example : differentiate y=e's" '

Ans :

Example : differentiate y=e's" '

Ans'

. y'

.

- e."""

C-sink))

Application : what about ¥Ea× ] for general

a > 0 ?

Note : ax = ( a )! (e. Inca' )×= elncaix

Chain rulesays

ddz[ax ] = date 'm"] = elncalx ( Inca) ) = of Inca)

We saw how to use product rule with more

than one product fcxlgcxlhlx) .

Chain rule with more chains?

y= (cul , u=gCx) ,x-

- htt)

htt -- fit # IF

Example : fcxl -- tanccoscsincxl )) ,

find f'

Ans :

Example : fat -- tanccoscsincxl ))

Ans'

. Xx ) -- gchcpcxl)) , gcxl -- tank)h (x ) - cos (x)

(Tx) ' g'Chcpcxil)hYpCxDpY×) pcxlesincx )

= see ( coscsincxll) C- sin (sink) )) ( cos Cx) )

Nothing special ! Think of flx) -- gcqlx)) . qcxl .- coscsinlxl)

( Tx )-

- g'Cqcx)) q'Cx) , now we just do chain rule

again to get Ctx)

try tanccoscxhti )) at home

Example : differentiate fca) =etanco)

Ans'

.

Example : differentiate econ) =etancoi)

Ans'

. foot-

- gchcpcoit) , gct) - ethCot= fault)

PCE) = -02

(Cfl - g'(hepcat) h'Cpcfl ) pyo)

= etanct')secco') (20)

Exam next week covers upto

AND including 3.4