# 4.3 Partial derivatives  (Page 7/11)

 Page 7 / 11

## 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 $x$
$\frac{\partial f}{\partial x}=\underset{h\to 0}{\text{lim}}\frac{f\left(x+h,y\right)-f\left(x,y\right)}{h}$
• Partial derivative of $f$ with respect to $y$
$\frac{\partial f}{\partial y}=\underset{k\to 0}{\text{lim}}\frac{f\left(x,y+k\right)-f\left(x,y\right)}{k}$

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

$\frac{\partial z}{\partial x}$ for $z={x}^{2}-3xy+{y}^{2}$

$\frac{\partial z}{\partial y}$ for $z={x}^{2}-3xy+{y}^{2}$

$\frac{\partial z}{\partial y}=-3x+2y$

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

${f}_{x}\left(1,1\right)$

${f}_{x}\left(-1,1\right)$

The sign is negative.

${f}_{y}\left(1,1\right)$

${f}_{x}\left(0,0\right)$

The partial derivative is zero at the origin.

For the following exercises, calculate the partial derivatives.

$\frac{\partial z}{\partial x}$ for $z=\text{sin}\left(3x\right)\text{cos}\left(3y\right)$

$\frac{\partial z}{\partial y}$ for $z=\text{sin}\left(3x\right)\text{cos}\left(3y\right)$

$\frac{\partial z}{\partial y}=-3\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3x\right)\text{sin}\left(3y\right)$

$\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z={x}^{8}{e}^{3y}$

$\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z=\text{ln}\left({x}^{6}+{y}^{4}\right)$

$\frac{\partial z}{\partial x}=\frac{6{x}^{5}}{{x}^{6}+{y}^{4}};\frac{\partial z}{\partial y}=\frac{4{y}^{3}}{{x}^{6}+{y}^{4}}$

Find ${f}_{y}\left(x,y\right)$ for $f\left(x,y\right)={e}^{xy}\text{cos}\left(x\right)\text{sin}\left(y\right).$

Let $z={e}^{xy}.$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$

$\frac{\partial z}{\partial x}=y{e}^{xy};\frac{\partial z}{\partial y}=x{e}^{xy}$

Let $z=\text{ln}\left(\frac{x}{y}\right).$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$

Let $z=\text{tan}\left(2x-y\right).$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$

$\frac{\partial z}{\partial x}=2\phantom{\rule{0.2em}{0ex}}{\text{sec}}^{2}\left(2x-y\right),\frac{\partial z}{\partial y}=\text{−}{\text{sec}}^{2}\left(2x-y\right)$

Let $z=\text{sinh}\left(2x+3y\right).$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}.$

Let $f\left(x,y\right)=\text{arctan}\left(\frac{y}{x}\right).$ Evaluate ${f}_{x}\left(2,-2\right)$ and ${f}_{y}\left(2,-2\right).$

${f}_{x}\left(2,-2\right)=\frac{1}{4}={f}_{y}\left(2,-2\right)$

Let $f\left(x,y\right)=\frac{xy}{x-y}.$ Find ${f}_{x}\left(2,-2\right)$ and ${f}_{y}\left(2,-2\right).$

Evaluate the partial derivatives at point $P\left(0,1\right).$

Find $\frac{\partial z}{\partial x}$ at $\left(0,1\right)$ for $z={e}^{\text{−}x}\text{cos}\left(y\right).$

$\frac{\partial z}{\partial x}=\text{−}\text{cos}\left(1\right)$

Given $f\left(x,y,z\right)={x}^{3}y{z}^{2},$ find $\frac{{\partial }^{2}f}{\partial x\partial y}$ and ${f}_{z}\left(1,1,1\right).$

Given $f\left(x,y,z\right)=2\phantom{\rule{0.2em}{0ex}}\text{sin}\left(x+y\right),$ find ${f}_{x}\left(0,\frac{\pi }{2},-4\right),$ ${f}_{y}\left(0,\frac{\pi }{2},-4\right),$ and ${f}_{z}\left(0,\frac{\pi }{2},-4\right).$

$\begin{array}{ccc}{f}_{x}=0,\hfill & {f}_{y}=0,\hfill & {f}_{z}=0\hfill \end{array}$

The area of a parallelogram with adjacent side lengths that are $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b,$ and in which the angle between these two sides is $\theta ,$ is given by the function $A\left(a,b,\theta \right)=ba\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\theta \right).$ Find the rate of change of the area of the parallelogram with respect to the following:

1. Side a
2. Side b
3. $\text{Angle}\phantom{\rule{0.2em}{0ex}}\theta$

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. $V\left(r,h\right)=\pi {r}^{2}h$ b. $\frac{\partial V}{\partial r}=2\pi rh$ c. $\frac{\partial V}{\partial h}=\pi {r}^{2}$

Calculate $\frac{\partial w}{\partial z}$ for $w=z\phantom{\rule{0.2em}{0ex}}\text{sin}\left(x{y}^{2}+2z\right).$

Find the indicated higher-order partial derivatives.

${f}_{xy}$ for $z=\text{ln}\left(x-y\right)$

${f}_{xy}=\frac{1}{{\left(x-y\right)}^{2}}$

${f}_{yx}$ for $z=\text{ln}\left(x-y\right)$

Let $z={x}^{2}+3xy+2{y}^{2}.$ Find $\frac{{\partial }^{2}z}{\partial {x}^{2}}$ and $\frac{{\partial }^{2}z}{\partial {y}^{2}}.$

$\frac{{\partial }^{2}z}{\partial {x}^{2}}=2,\frac{{\partial }^{2}z}{\partial {y}^{2}}=4$

Given $z={e}^{x}\text{tan}\phantom{\rule{0.2em}{0ex}}y,$ find $\frac{{\partial }^{2}z}{\partial x\partial y}$ and $\frac{{\partial }^{2}z}{\partial y\partial x}.$

Given $f\left(x,y,z\right)=xyz,$ find ${f}_{xyy},{f}_{yxy},$ and ${f}_{yyx}.$

${f}_{xyy}={f}_{yxy}={f}_{yyx}=0$

Given $f\left(x,y,z\right)={e}^{-2x}\text{sin}\left({z}^{2}y\right),$ show that ${f}_{xyy}={f}_{yxy}.$

Show that $z=\frac{1}{2}\left({e}^{y}-{e}^{\text{−}y}\right)\text{sin}\phantom{\rule{0.2em}{0ex}}x$ is a solution of the differential equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=0.$

$\begin{array}{ccc}\hfill \frac{{d}^{2}z}{d{x}^{2}}& =\hfill & -\frac{1}{2}\left({e}^{y}-{e}^{\text{−}y}\right)\text{sin}\phantom{\rule{0.2em}{0ex}}x\hfill \\ \hfill \frac{{d}^{2}z}{d{y}^{2}}& =\hfill & \frac{1}{2}\left({e}^{y}-{e}^{\text{−}y}\right)\text{sin}\phantom{\rule{0.2em}{0ex}}x\hfill \\ \hfill \frac{{d}^{2}z}{d{x}^{2}}+\frac{{d}^{2}z}{d{y}^{2}}& =\hfill & 0\hfill \end{array}$

Find ${f}_{xx}\left(x,y\right)$ for $f\left(x,y\right)=\frac{4{x}^{2}}{y}+\frac{{y}^{2}}{2x}.$

Let $f\left(x,y,z\right)={x}^{2}{y}^{3}z-3x{y}^{2}{z}^{3}+5{x}^{2}z-{y}^{3}z.$ Find ${f}_{xyz}.$

${f}_{xyz}=6{y}^{2}x-18y{z}^{2}$

Let $F\left(x,y,z\right)={x}^{3}y{z}^{2}-2{x}^{2}yz+3xz-2{y}^{3}z.$ Find ${F}_{xyz}.$

Given $f\left(x,y\right)={x}^{2}+x-3xy+{y}^{3}-5,$ find all points at which ${f}_{x}={f}_{y}=0$ simultaneously.

$\left(\frac{1}{4},\frac{1}{2}\right),\left(1,1\right)$

Given $f\left(x,y\right)=2{x}^{2}+2xy+{y}^{2}+2x-3,$ find all points at which $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$ simultaneously.

Given $f\left(x,y\right)={y}^{3}-3y{x}^{2}-3{y}^{2}-3{x}^{2}+1,$ find all points on $f$ at which ${f}_{x}={f}_{y}=0$ simultaneously.

$\left(0,0\right),\left(0,2\right),\left(\sqrt{3},-1\right),\left(\text{−}\sqrt{3},-1\right)$

Given $f\left(x,y\right)=15{x}^{3}-3xy+15{y}^{3},$ find all points at which ${f}_{x}\left(x,y\right)={f}_{y}\left(x,y\right)=0$ simultaneously.

Show that $z={e}^{x}\text{sin}\phantom{\rule{0.2em}{0ex}}y$ satisfies the equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=0.$

$\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}={e}^{x}\text{sin}\left(y\right)-{e}^{x}\text{sin}\phantom{\rule{0.2em}{0ex}}y=0$

Show that $f\left(x,y\right)=\text{ln}\left({x}^{2}+{y}^{2}\right)$ solves Laplace’s equation $\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=0.$

Show that $z={e}^{\text{−}t}\text{cos}\left(\frac{x}{c}\right)$ satisfies the heat equation $\frac{\partial z}{\partial t}=\text{−}{e}^{\text{−}t}\text{cos}\left(\frac{x}{c}\right).$

${c}^{2}\frac{{\partial }^{2}z}{\partial {x}^{2}}={e}^{\text{−}t}\text{cos}\left(\frac{x}{c}\right)$

Find $\underset{\text{Δ}x\to 0}{\text{lim}}\frac{f\left(x+\text{Δ}x\right)-f\left(x,y\right)}{\text{Δ}x}$ for $f\left(x,y\right)=-7x-2xy+7y.$

Find $\underset{\text{Δ}y\to 0}{\text{lim}}\frac{f\left(x,y+\text{Δ}y\right)-f\left(x,y\right)}{\text{Δ}y}$ for $f\left(x,y\right)=-7x-2xy+7y.$

$\frac{\partial f}{\partial y}=-2x+7$

Find $\underset{\text{Δ}x\to 0}{\text{lim}}\frac{\text{Δ}f}{\text{Δ}x}=\underset{\text{Δ}x\to 0}{\text{lim}}\frac{f\left(x+\text{Δ}x,y\right)-f\left(x,y\right)}{\text{Δ}x}$ for $f\left(x,y\right)={x}^{2}{y}^{2}+xy+y.$

Find $\underset{\text{Δ}x\to 0}{\text{lim}}\frac{\text{Δ}f}{\text{Δ}x}=\underset{\text{Δ}x\to 0}{\text{lim}}\frac{f\left(x+\text{Δ}x,y\right)-f\left(x,y\right)}{\text{Δ}x}$ for $f\left(x,y\right)=\text{sin}\left(xy\right).$

$\frac{\partial f}{\partial x}=y\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}xy$

The function $P\left(T,V\right)=\frac{nRT}{V}$ gives the pressure at a point in a gas as a function of temperature $T$ and volume $V.$ The letters $n\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R$ are constants. Find $\frac{\partial P}{\partial V}$ and $\frac{\partial P}{\partial T},$ and explain what these quantities represent.

The equation for heat flow in the $xy\text{-plane}$ is $\frac{\partial f}{\partial t}=\frac{{\partial }^{2}f}{\partial {x}^{2}}+\frac{{\partial }^{2}f}{\partial {y}^{2}}.$ Show that $f\left(x,y,t\right)={e}^{-2t}\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y$ is a solution.

The basic wave equation is ${f}_{tt}={f}_{xx}.$ Verify that $f\left(x,t\right)=\text{sin}\left(x+t\right)$ and $f\left(x,t\right)=\text{sin}\left(x-t\right)$ are solutions.

The law of cosines can be thought of as a function of three variables. Let $x,y,$ and $\theta$ be two sides of any triangle where the angle $\theta$ is the included angle between the two sides. Then, $F\left(x,y,\theta \right)={x}^{2}+{y}^{2}-2xy\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$ gives the square of the third side of the triangle. Find $\frac{\partial F}{\partial \theta }$ and $\frac{\partial F}{\partial x}$ when $x=2,y=3,$ and $\theta =\frac{\pi }{6}.$

$\frac{\partial F}{\partial \theta }=6,\frac{\partial F}{\partial x}=4-3\sqrt{3}$

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.)

A Cobb-Douglas production function is $f\left(x,y\right)=200{x}^{0.7}{y}^{0.3},$ where $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y$ represent the amount of labor and capital available. Let $x=500$ and $y=1000.$ Find $\frac{\delta f}{\delta x}$ and $\frac{\delta f}{\delta y}$ at these values, which represent the marginal productivity of labor and capital, respectively.

$\frac{\delta f}{\delta x}$ at $\left(500,1000\right)=172.36,$ $\frac{\delta f}{\delta y}$ at $\left(500,1000\right)=36.93$

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.

$A=0.885t-22.4h+1.20th-0.544.$ Find $\frac{\partial A}{\partial t}$ and $\frac{\partial A}{\partial h}$ when $t=20\text{°}\text{F}$ and $h=0.90.$

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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