# 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 $h=0.90.$

#### Questions & Answers

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!