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