<< Chapter < Page Chapter >> Page >
  • State the chain rules for one or two independent variables.
  • Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables.
  • Perform implicit differentiation of a function of two or more variables.

In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. In this section, we study extensions of the chain rule and learn how to take derivatives of compositions of functions of more than one variable.

Chain rules for one or two independent variables

Recall that the chain rule for the derivative of a composite of two functions can be written in the form

d d x ( f ( g ( x ) ) ) = f ( g ( x ) ) g ( x ) .

In this equation, both f ( x ) and g ( x ) are functions of one variable. Now suppose that f is a function of two variables and g is a function of one variable. Or perhaps they are both functions of two variables, or even more. How would we calculate the derivative in these cases? The following theorem gives us the answer for the case of one independent variable.

Chain rule for one independent variable

Suppose that x = g ( t ) and y = h ( t ) are differentiable functions of t and z = f ( x , y ) is a differentiable function of x and y . Then z = f ( x ( t ) , y ( t ) ) is a differentiable function of t and

d z d t = z x · d x d t + z y · d y d t ,

where the ordinary derivatives are evaluated at t and the partial derivatives are evaluated at ( x , y ) .

Proof

The proof of this theorem uses the definition of differentiability of a function of two variables. Suppose that f is differentiable at the point P ( x 0 , y 0 ) , where x 0 = g ( t 0 ) and y 0 = h ( t 0 ) for a fixed value of t 0 . We wish to prove that z = f ( x ( t ) , y ( t ) ) is differentiable at t = t 0 and that [link] holds at that point as well.

Since f is differentiable at P , we know that

z ( t ) = f ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) + E ( x , y ) ,

where lim ( x , y ) ( x 0 , y 0 ) E ( x , y ) ( x x 0 ) 2 + ( y y 0 ) 2 = 0 . We then subtract z 0 = f ( x 0 , y 0 ) from both sides of this equation:

z ( t ) z ( t 0 ) = f ( x ( t ) , y ( t ) ) f ( x ( t 0 ) , y ( t 0 ) ) = f x ( x 0 , y 0 ) ( x ( t ) x ( t 0 ) ) + f y ( x 0 , y 0 ) ( y ( t ) y ( t 0 ) ) + E ( x ( t ) , y ( t ) ) .

Next, we divide both sides by t t 0 :

z ( t ) z ( t 0 ) t t 0 = f x ( x 0 , y 0 ) ( x ( t ) x ( t 0 ) t t 0 ) + f y ( x 0 , y 0 ) ( y ( t ) y ( t 0 ) t t 0 ) + E ( x ( t ) , y ( t ) ) t t 0 .

Then we take the limit as t approaches t 0 :

lim t t 0 z ( t ) z ( t 0 ) t t 0 = f x ( x 0 , y 0 ) lim t t 0 ( x ( t ) x ( t 0 ) t t 0 ) + f y ( x 0 , y 0 ) lim t t 0 ( y ( t ) y ( t 0 ) t t 0 ) + lim t t 0 E ( x ( t ) , y ( t ) ) t t 0 .

The left-hand side of this equation is equal to d z / d t , which leads to

d z d t = f x ( x 0 , y 0 ) d x d t + f y ( x 0 , y 0 ) d y d t + lim t t 0 E ( x ( t ) , y ( t ) ) t t 0 .

The last term can be rewritten as

lim t t 0 E ( x ( t ) , y ( t ) ) t t 0 = lim t t 0 ( E ( x , y ) ( x x 0 ) 2 + ( y y 0 ) 2 ( x x 0 ) 2 + ( y y 0 ) 2 t t 0 ) = lim t t 0 ( E ( x , y ) ( x x 0 ) 2 + ( y y 0 ) 2 ) lim t t 0 ( ( x x 0 ) 2 + ( y y 0 ) 2 t t 0 ) .

As t approaches t 0 , ( x ( t ) , y ( t ) ) approaches ( x ( t 0 ) , y ( t 0 ) ) , so we can rewrite the last product as

lim ( x , y ) ( x 0 , y 0 ) ( E ( x , y ) ( x x 0 ) 2 + ( y y 0 ) 2 ) lim ( x , y ) ( x 0 , y 0 ) ( ( x x 0 ) 2 + ( y y 0 ) 2 t t 0 ) .

Since the first limit is equal to zero, we need only show that the second limit is finite:

lim ( x , y ) ( x 0 , y 0 ) ( ( x x 0 ) 2 + ( y y 0 ) 2 t t 0 ) = lim ( x , y ) ( x 0 , y 0 ) ( ( x x 0 ) 2 + ( y y 0 ) 2 ( t t 0 ) 2 ) = lim ( x , y ) ( x 0 , y 0 ) ( ( x x 0 t t 0 ) 2 + ( y y 0 t t 0 ) 2 ) = ( lim ( x , y ) ( x 0 , y 0 ) ( x x 0 t t 0 ) ) 2 + ( lim ( x , y ) ( x 0 , y 0 ) ( y y 0 t t 0 ) ) 2 .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask