The Chain Rule

Independent variableRecally Let y 7 x

dat Instantaneous rate of change of ywith respect to K

Q what happens when y can be expressed in

terms of a different independent variable

MotivalingtA leaking oil well is spreading a c film

over a water surface At time t after the

start of the Leak the radius is 4T

Let A surface area of spill r radius of

spill and t time after start of leak

It can be a Function in r e t Is there

d A d Aa relationship between and Taear

A Cr Itv2

Act IT ret IT 4 72 16 EZ

dat ZIT r ariddA 32 Itat

Observationsd A 2 r ZIT t 81T td r

d r

Tt It Gt 4

dd 32Mt SITE 4d

drat

conclusion

Rate of Change Rate ARate of change7 A with Changest A of r with

respect to t with respect respect toto r t

Chain Rule Form 1

y 7 u y is a function in u

U g x c u is a Function in t

dy dy du Clever NotationNot really cancellationd c du DX of fractions

Example1 y Tu dyu z fan Gc

If Facial _Eu E

P Rdu

dyZu z 2 Gaul c t 2 seckx

E ETru Lutz 2 Eaulx t 2 seckx

Need toswitchto x Lutz 2 taulx t 2 seckx

2 fair CK

z y Sec udy

u Casca I

dy tan u seccudu

duTx

Siu ca

dy fan Cut seclu C Siu x

d Ctan Casca Sec cos x sin xD

Prime Notation

y fluY fCgc

composition

dyu g Csc In f u 7 g ca

duDI g Ge

chainRuleCF.ru

d

dy p 7cg ll 7 Cgc.ci gdy T du g

DX du dk

Conclusion The Chain Rule tells us how to

differentiate compositions of functions

Zemarky

Y You can use either form The advantage of

Liebuiz notation is that it explicitly tells us

the relevant independent variable For example is

dyy equal to Dor date

3 We can extend the Chain Rule to compositions ofmore functions For example

y 7 u

u gas dat dat dat datk h t

IdTt 7cgChat 7 Cgcuct g net h Ct

Exampley d since

af 2

y z

u Sin x date tu z 2 cos Cas et

Siu t2 cos 1 2 2Ta C 2 Tu z 2

Useful Standard Examples

Y 7 Cmx b 7 math m

Y da b9 Inch b9 g ex

7 genr

regex g ex

Product Rate Chain Rule Quotient Rule

tg ce cai.cgc.cn I

text cgcxi5 text Cgc as 5

7 x

gett 7cal

ca z g ca

7 d gca 7 x g x

go.sc2