What is Partial differentiation: Definition and 126 Discussions
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The partial derivative of a function
f
(
x
,
y
,
…
)
{\displaystyle f(x,y,\dots )}
with respect to the variable
x
{\displaystyle x}
is variously denoted by
f
x
′
,
f
x
,
∂
x
f
,
D
x
f
,
D
1
f
,
∂
∂
x
f
,
or
∂
f
∂
x
.
{\displaystyle f'_{x},f_{x},\partial _{x}f,\ D_{x}f,D_{1}f,{\frac {\partial }{\partial x}}f,{\text{ or }}{\frac {\partial f}{\partial x}}.}
Sometimes, for
z
=
f
(
x
,
y
,
…
)
,
{\displaystyle z=f(x,y,\ldots ),}
the partial derivative of
z
{\displaystyle z}
with respect to
x
{\displaystyle x}
is denoted as
∂
z
∂
x
.
{\displaystyle {\tfrac {\partial z}{\partial x}}.}
Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
f
x
(
x
,
y
,
…
)
,
∂
f
∂
x
(
x
,
y
,
…
)
.
{\displaystyle f_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).}
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786) (although he later abandoned it, Carl Gustav Jacob Jacobi reintroduced the symbol in 1841).
Hi;
please see below I am trying to understand how to get to the 2 final functions. They should be the same but 6 for the first one and 2 for the second?
(I hope my writing is more clear than previously)
There is an additional question below.
thanks
martyn
I can't find a standard derivative...
Attempt at question No. 1:
ΔD = ∂D/∂h * Δh + ∂D/∂v * Δv
∂D/∂h = 3Eh^2/(12(1-v^2))
∂D/∂v = 2Eh^3/(12(1-v^2)^2)
Δh = +- 0,002
Δv = 0,02
h = 0,1
v = 0,3
ΔD = 3Eh^2/(12(1-v^2)) * Δh + 2Eh^3/(12(1-v^2)^2) * Δv
Because the problem asked for maximum percentage error then I decided to use the...
Hi
For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ?
For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ?
Thanks
$$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$
They wrote that above equation is equal to...
𝝏w/𝝏x=1
and then I wasn't sure about 𝝏x/𝝏s, so I tried implicitly differentiating s:
1=(3x^2)(𝝏x/𝝏s)+y(𝝏x/𝝏s)+x(𝝏y/𝝏s)+(3y^2)(𝝏y/𝝏s)
And then I shaved my head in frustration.
zx = 2xy + y2 -3y = 0 and zy = 2xy + x2 - 3x = 0
Subtracting one equation from the other gives
y2 - 3y = x2- 3x ⇒ y (y-3) = x (x-3)
This leads to the following solutions ( 0 , 0) , (0 ,3) , (3 , 0) but the answer also gives ( 1, 1) as a solution. What have i done wrong to not get this...
In Section 7.6 - Equivalence of Lagrange's and Newton's Equations in the Classical Dynamics of Particles and Systems book by Thornton and Marion, pages 255 and 256, introduces the following transformation from the xi-coordinates to the generalized coordinates qj in Equation (7.99):
My...
1. The problem statement, all variables, and given/known data
Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##.
Homework Equations
Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...
Why we write differently d in partial derivation differentiation? Is it because of several variables?
Edited by mentor -- the action of finding a derivative is called differentiation.
I have a very basic knowledge of calculus of one variable .
In the chapter on heat and thermodynamics , ideal gas law PV =nRT is given .
Then the book says, differentiating you get
PdV +VdP = nRdT .
The book doesn't explain the differentiation step .
I think , there are two ways to...
I am working my way through a textbook, and whenever this equation is solved (integrated), the answer is given as:
u = f(x) + f(y)
I don't understand it. If I integrate it once (with respect to y, say), then I obtain:
∂u/∂x = f(x) -----eq.1
If I integrate again (this time with respect to...
Homework Statement
Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions:
\begin{equation}
\frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0
\end{equation}
\begin{equation}
u(x,0) = cos(x)...
I believe there is a mistake in the second equation of (5.139).
The equation is obtained from (5.138) using the Euler-Lagrange equation
##\frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial\theta}.##
LHS##\,\,=\frac{d}{dt}\frac{\partial...
Homework Statement
in the notes , 'by applying chain rule to LHS of the above equation ' , which equation is the author referring to ?
it's given that
f /x + (f/z)(z/x) = 0 ,
As we can see , the function contain variable x , y and z
Homework EquationsThe Attempt at a Solution
why not
f /x +...
Hello guys,
I'm doing my physics coursework on kepler's third law and I'm finding the minimum mass and semi-major axis of a unknown planet. I have the following data:
Stellar mass Mstar = 1.31 ± 0.05 Msun
Orbital period P = 2.243752 ± 0.00005 days
Radial velocity semi-amplitude: V = 993.0 ±...
Hi, in the above why is the left-hand side simple differentiation, i.e V is only function of t but in the right it is function of t, x, y, and z. It is very strange that one side is different than the other. Would you like to explain it?
Thank you.
Hey all,
I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation:
\frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...
If I have a function
##f(u,u^*) = \int u^* \hat{O} u d^3\mathbf{r}##
both ##u## and ##u^*## are functions of ##\mathbf{r}## where ##\mathbf{r}## position vector, ##\hat{O}## some operation which involves ##\mathbf{r}## (e.g. differentiation), and the star sign denotes complex conjugate. Now I...
Homework Statement
Suppose we have an equation,
ex + xy + x2 = 5
Find dy/dx
Homework Equations
Now I know all the linear differentiation stuff like product rule, chain rule etc.
Also I know partial differentiation is differentiating one variable and keeping other one constant.
The Attempt at...
Homework Statement
Given: z = f(x,y) = x^2-y^2
To take the partial derivative of f with respect to x hold y constant then take the derivative of x.
∂f/∂x = 2x
What I don't understand is why such would equal 2x, when the y is still there it just isn't variable and is ignored. Wouldn't it be...
Homework Statement
Find \frac{\partial}{\partial x} if:
f(x,y) = \begin{cases}x^2\frac{\sin y}{y}, & y\neq 0\\0, &y=0 \end{cases}
Homework EquationsThe Attempt at a Solution
If y\neq 0 , then it's simple, but I get confused about the second part. How can I exactly utilize the limit definition...
Homework Statement
The problem and my attempt are attached
Homework Equations
Chain rule for partial differentiation perhaps
And basic algebra
The Attempt at a Solution
I'm unsure of how to approach this but I differentiated all the expression at the top.
Homework Statement
if z=\frac{1}{x^2+y^2-1} . Show that x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = -2z(1+z)
Homework Equations
n/a
The Attempt at a Solution
I am extremely new to partial differentiation, I can get my head around questions where they just give...
Homework Statement
if z = x2 + 2y2 , x = r cos θ , y = r sin θ , find the partial derivative
\left(\frac{\partial z}{\partial \theta}\right)_{x}
Homework Equations
z = x2 + 2y2
x = r cos θ
y = r sin θ
The Attempt at a Solution
The textbook says that the equation should be...
I am not quite sure how \frac{\partial}{\partial u}\left(\frac{\partial z}{\partial u}\right)
=\frac{\partial}{\partial u}\left( u \frac{\partial z}{\partial x}+v\frac{\partial z}{\partial y} \right)
comes to \frac{\partial z}{\partial x} + u\frac{\partial}{\partial u}\left(\frac{\partial...
Hello MHB members and friends!(Callme)
An economy student asked me, if I could explain the following partial differentiation:
\[\frac{\partial}{\partial C(i)}\int_{i\in[0;1]}[C(i)]^\frac{\eta - 1}{\eta}di
=\int_{j\in[0;1]}[C(j)]^\frac{\eta - 1}{\eta}dj\frac{\eta -...
Homework Statement
Show that a relation of the kind ƒ(x,y,z) = 0
then implies the relation
(∂x/∂y)_z (∂y/∂z)_x (∂z/∂x)_y = -1
Homework Equations
f(x,y)
df = (∂f/∂x)_y dx + (∂f/∂y)_x dy
The Attempt at a Solution
I expressed x = x(y,z) and y = y(x,z) then found dx and...
Homework Statement
z = x^2 +y^2
x = rcosθ
y = rsinθ
find partial z over partial x at constant theta
Homework Equations
z = x^2 +y^2
x = rcosθ
y = rsinθ
The Attempt at a Solution
z = 1 + r^2(sinθ)^2
dz/dx = dz/dr . dr/dx
= 2(sinθ)^2r/cosθ
= 2tanθ^2x...
Homework Statement
let V=f(x²+y²) , show that x(∂V/∂y) - y(∂V/∂x) = 0
Homework Equations
The Attempt at a Solution
V=f(x²+y²) ; V=f(x)² + f(y)²
∂V/∂x = 2[f(x)]f'(x) + [0]
∂V/∂y = 2[f(y)]f'(y)
I'm sure I've gone wrong somewhere, I have never seen functions like this...
I got x = (u2 - v2) / u
y = (v2 - u2) / v
I differentiated them w.r.t u & v respectively & solved the given equation but I'm not getting the answer which is 0.
Please view attachment for question!
Given a function: z(x,y) = 2x +2y^2
Determine ∂x/∂y [the partial differentiation of x with respect to y],
Method 1:
x = (z/2) - y^2
∂x/∂y = -2y
Method 2:
∂z/∂x = 2
∂z/∂y = 4y
∂x/∂y = ∂x/∂z X ∂z/∂y = (1/2) X 4y = 2y
One or both of these is wrong. Can someone point out...
I just want to verify
For Polar coordinates, ##r^2=x^2+y^2## and ##x=r\cos \theta##, ##y=r\sin\theta##
##x(r,\theta)## and## y(r,\theta)## are not independent to each other like in rectangular.
In rectangular coordinates, ##\frac{\partial y}{\partial x}=\frac{dy}{dx}=0##
But in Polar...
Homework Statement
If ##z=x\ln(x+r)-r## where ##r^2=x^2+y^2##, prove that
$$\frac{∂^2z}{∂x^2}+\frac{∂^2z}{∂y^2}=\frac{1}{x+y}$$Homework Equations
The Attempt at a Solution
Since ##r^2=x^2+y^2##, ##∂r/∂x=x/r## and ##∂r/∂y=y/r##.
Differentiating z w.r.t x partially...
Hello,
the question I have arises from the 4th Edition of the book "Advanced Engineering Mathematics" written by K.A. Stroud. For those who owns the book, it is the example #2 starting at page 379. More precisely, the example is separated into two parts but the first one is very straight...
Homework Statement
I found this solved example in an old textbook. I don't think that the solution provided is correct. I'll be very grateful if someone could verify it.
Question:
xxyyzz = c
What is \frac{∂z}{∂x}?
Solution Provided:
Taking logarithms on both sides:
zlog(z) =...
Homework Statement
y(x,t) = f(x-ct)
verify this solution satisfies equation
∂y2/∂x2 = 1/c2*∂y2/∂t2
Homework Equations
The Attempt at a Solution
∂y/∂x = ∂f/∂x = 1
∂y2/∂x2 = 0
∂y/∂t = ∂f/∂t = -c
∂y2/∂t2 = 0
Is this the way to do it?
Homework Statement
The function f(x,y,z) may be expressed in new coordinates as g(u,v,w). Prove this general result:
The Attempt at a Solution
df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz
dg = (∂g/∂u)du + (∂g/∂v)dv + (∂g/∂w)dw
df = dg since they are the same thing?
but the...
Homework Statement
Consider the following equality:
(\frac{∂S}{∂V})T = (\frac{∂P}{∂T})V
If I rearrange the equality so that I write:
(\frac{∂S}{∂P})? = (\frac{∂V}{∂T})?
What variables will be constant in each side?
I'm having some trouble in a few thermodynamics problems because...
Homework Statement
Calculate ∂f/∂x and ∂f/∂y for the following function:
yf^2 + sin(xy) = f
The Attempt at a Solution
I understand basic partial differentiation, but I have no idea how to approach the f incorporation on both sides of the equation nor what you would explicitly call this...
Folks,
I am stuck on an example which is partial differenting a functional with indicial notation
The functional ##\displaystyle I(c_1,c_2,...c_N)=\frac{1}{2} \int_0^1 \left [ \left (\sum\limits_{j=1}^N c_j \frac{d \phi_j}{dx}\right )^2-\left(\sum\limits_{j=1}^N c_j \phi_j\right)^2+2x^2...
Homework Statement
This question is about entropy of magnetic salts. I got up to the point of finding H1, the final applied field.
The Attempt at a Solution
But instead of doing integration I used this:
dS = (∂S/∂H)*dH
= (M0/4α)(ln 4)2
I removed the negative...
Hi everyone,
I know that if
z = f(x,y) = x^2y + xy^2
then
\frac{\partial z}{\partial x}=2xy+y^2 and
\frac{\partial z}{\partial y}=x^2+2xy
Please correct me if I am wrong.
In the physics, can anyone please tell me what is the meaning of below formula?
\frac{\partial V}{\partial t}
Where...