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Key concepts

  • A partial derivative is a derivative involving a function of more than one independent variable.
  • To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.
  • Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.

Key equations

  • Partial derivative of f with respect to x
    f x = lim h 0 f ( x + h , y ) f ( x , y ) h
  • Partial derivative of f with respect to y
    f y = lim k 0 f ( x , y + k ) f ( x , y ) k

For the following exercises, calculate the partial derivative using the limit definitions only.

z x for z = x 2 3 x y + y 2

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z y for z = x 2 3 x y + y 2

z y = −3 x + 2 y

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For the following exercises, calculate the sign of the partial derivative using the graph of the surface.

A partial paraboloid with vertex at the origin and pointing up.

f x ( −1 , 1 )

The sign is negative.

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f x ( 0 , 0 )

The partial derivative is zero at the origin.

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For the following exercises, calculate the partial derivatives.

z x for z = sin ( 3 x ) cos ( 3 y )

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z y for z = sin ( 3 x ) cos ( 3 y )

z y = −3 sin ( 3 x ) sin ( 3 y )

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z x and z y for z = x 8 e 3 y

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z x and z y for z = ln ( x 6 + y 4 )

z x = 6 x 5 x 6 + y 4 ; z y = 4 y 3 x 6 + y 4

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Find f y ( x , y ) for f ( x , y ) = e x y cos ( x ) sin ( y ) .

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Let z = e x y . Find z x and z y .

z x = y e x y ; z y = x e x y

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Let z = ln ( x y ) . Find z x and z y .

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Let z = tan ( 2 x y ) . Find z x and z y .

z x = 2 sec 2 ( 2 x y ) , z y = sec 2 ( 2 x y )

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Let z = sinh ( 2 x + 3 y ) . Find z x and z y .

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Let f ( x , y ) = arctan ( y x ) . Evaluate f x ( 2 , −2 ) and f y ( 2 , −2 ) .

f x ( 2 , −2 ) = 1 4 = f y ( 2 , −2 )

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Let f ( x , y ) = x y x y . Find f x ( 2 , −2 ) and f y ( 2 , −2 ) .

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Evaluate the partial derivatives at point P ( 0 , 1 ) .

Find z x at ( 0 , 1 ) for z = e x cos ( y ) .

z x = cos ( 1 )

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Given f ( x , y , z ) = x 3 y z 2 , find 2 f x y and f z ( 1 , 1 , 1 ) .

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Given f ( x , y , z ) = 2 sin ( x + y ) , find f x ( 0 , π 2 , −4 ) , f y ( 0 , π 2 , −4 ) , and f z ( 0 , π 2 , −4 ) .

f x = 0 , f y = 0 , f z = 0

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The area of a parallelogram with adjacent side lengths that are a and b , and in which the angle between these two sides is θ , is given by the function A ( a , b , θ ) = b a sin ( θ ) . Find the rate of change of the area of the parallelogram with respect to the following:

  1. Side a
  2. Side b
  3. Angle θ
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Express the volume of a right circular cylinder as a function of two variables:

  1. its radius r and its height h .
  2. Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height.
  3. Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.

a. V ( r , h ) = π r 2 h b. V r = 2 π r h c. V h = π r 2

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Calculate w z for w = z sin ( x y 2 + 2 z ) .

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Find the indicated higher-order partial derivatives.

f x y for z = ln ( x y )

f x y = 1 ( x y ) 2

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f y x for z = ln ( x y )

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Let z = x 2 + 3 x y + 2 y 2 . Find 2 z x 2 and 2 z y 2 .

2 z x 2 = 2 , 2 z y 2 = 4

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Given z = e x tan y , find 2 z x y and 2 z y x .

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Given f ( x , y , z ) = x y z , find f x y y , f y x y , and f y y x .

f x y y = f y x y = f y y x = 0

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Given f ( x , y , z ) = e −2 x sin ( z 2 y ) , show that f x y y = f y x y .

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Show that z = 1 2 ( e y e y ) sin x is a solution of the differential equation 2 z x 2 + 2 z y 2 = 0 .

d 2 z d x 2 = 1 2 ( e y e y ) sin x d 2 z d y 2 = 1 2 ( e y e y ) sin x d 2 z d x 2 + d 2 z d y 2 = 0

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Find f x x ( x , y ) for f ( x , y ) = 4 x 2 y + y 2 2 x .

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Let f ( x , y , z ) = x 2 y 3 z 3 x y 2 z 3 + 5 x 2 z y 3 z . Find f x y z .

f x y z = 6 y 2 x 18 y z 2

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Let F ( x , y , z ) = x 3 y z 2 2 x 2 y z + 3 x z 2 y 3 z . Find F x y z .

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Given f ( x , y ) = x 2 + x 3 x y + y 3 5 , find all points at which f x = f y = 0 simultaneously.

( 1 4 , 1 2 ) , ( 1 , 1 )

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Given f ( x , y ) = 2 x 2 + 2 x y + y 2 + 2 x 3 , find all points at which f x = 0 and f y = 0 simultaneously.

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Given f ( x , y ) = y 3 3 y x 2 3 y 2 3 x 2 + 1 , find all points on f at which f x = f y = 0 simultaneously.

( 0 , 0 ) , ( 0 , 2 ) , ( 3 , −1 ) , ( 3 , −1 )

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Given f ( x , y ) = 15 x 3 3 x y + 15 y 3 , find all points at which f x ( x , y ) = f y ( x , y ) = 0 simultaneously.

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Show that z = e x sin y satisfies the equation 2 z x 2 + 2 z y 2 = 0 .

2 z x 2 + 2 z y 2 = e x sin ( y ) e x sin y = 0

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Show that f ( x , y ) = ln ( x 2 + y 2 ) solves Laplace’s equation 2 z x 2 + 2 z y 2 = 0 .

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Show that z = e t cos ( x c ) satisfies the heat equation z t = e t cos ( x c ) .

c 2 2 z x 2 = e t cos ( x c )

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Find lim Δ x 0 f ( x + Δ x ) f ( x , y ) Δ x for f ( x , y ) = −7 x 2 x y + 7 y .

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Find lim Δ y 0 f ( x , y + Δ y ) f ( x , y ) Δ y for f ( x , y ) = −7 x 2 x y + 7 y .

f y = −2 x + 7

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Find lim Δ x 0 Δ f Δ x = lim Δ x 0 f ( x + Δ x , y ) f ( x , y ) Δ x for f ( x , y ) = x 2 y 2 + x y + y .

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Find lim Δ x 0 Δ f Δ x = lim Δ x 0 f ( x + Δ x , y ) f ( x , y ) Δ x for f ( x , y ) = sin ( x y ) .

f x = y cos x y

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The function P ( T , V ) = n R T V gives the pressure at a point in a gas as a function of temperature T and volume V . The letters n and R are constants. Find P V and P T , and explain what these quantities represent.

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The equation for heat flow in the x y -plane is f t = 2 f x 2 + 2 f y 2 . Show that f ( x , y , t ) = e −2 t sin x sin y is a solution.

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The basic wave equation is f t t = f x x . Verify that f ( x , t ) = sin ( x + t ) and f ( x , t ) = sin ( x t ) are solutions.

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The law of cosines can be thought of as a function of three variables. Let x , y , and θ be two sides of any triangle where the angle θ is the included angle between the two sides. Then, F ( x , y , θ ) = x 2 + y 2 2 x y cos θ gives the square of the third side of the triangle. Find F θ and F x when x = 2 , y = 3 , and θ = π 6 .

F θ = 6 , F x = 4 3 3

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Suppose the sides of a rectangle are changing with respect to time. The first side is changing at a rate of 2 in./sec whereas the second side is changing at the rate of 4 in/sec. How fast is the diagonal of the rectangle changing when the first side measures 16 in. and the second side measures 20 in.? (Round answer to three decimal places.)

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A Cobb-Douglas production function is f ( x , y ) = 200 x 0.7 y 0.3 , where x and y represent the amount of labor and capital available. Let x = 500 and y = 1000 . Find δ f δ x and δ f δ y at these values, which represent the marginal productivity of labor and capital, respectively.

δ f δ x at ( 500 , 1000 ) = 172.36 , δ f δ y at ( 500 , 1000 ) = 36.93

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The apparent temperature index is a measure of how the temperature feels, and it is based on two variables: h , which is relative humidity, and t , which is the air temperature.

A = 0.885 t 22.4 h + 1.20 t h 0.544 . Find A t and A h when t = 20 ° F and h = 0.90 .

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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..
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Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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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
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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?
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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
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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 4

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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