Partial derivative Definition and 374 Threads

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).

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  1. W

    I Partial derivative of Dirac delta of a composite argument

    I'm trying to prove the following statement: $$ D\partial_t\left(\delta\circ\mathbf{v}\right) = J^i\partial_i\left(\delta\circ\mathbf{v}\right), $$ where ##\mathbf{v}## is some function of time and ##n##-dimensional space, ## D ## is the Jacobian determinant associated with ##\mathbf{v}##, that...
  2. L

    I Differentiability of a Multivariable function

    I’m having a little confusion about part b of this question as to why I am allowed to use the limit definition of a partial derivative. Here’s what I think: I know that y^3/(x^2+y^2) is undefined at the origin but it does approach 0 when it GETS CLOSE to the origin. So technically defining...
  3. nrsakinh

    Need a real life example where a partial derivative is used in motion

    my group is preferring the ue of partial derivative to find the acceleration of a car or the projectile motion of something being launched
  4. workhorse123

    Potential in the three regions of an infinite slab

    for the boundary conditions for this problem I understand that Electric field and Electric potential will be continuous on the boundaries. I also know that I can set the reference point for Electric potential, wherever it is convenient. This gives me the fifth boundary condition. I am confused...
  5. S

    Find f(x,y) given partial derivative and initial condition

    My attempt: $$\frac{\partial f}{\partial x}=-\sin y + \frac{1}{1-xy}$$ $$\int \partial f=\int (-\sin y+\frac{1}{1-xy})\partial x$$ $$f=-x~\sin y-\frac{1}{y} \ln |1-xy|+c$$ Using ##f(0, y)=2 \sin y + y^3##: $$c=2 \sin y + y^3$$ So: $$f(x,y)=-x~\sin y-\frac{1}{y} \ln |1-xy|+2 \sin y + y^3$$ Is...
  6. B

    Partial Derivative Simplification

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  7. S

    Multivariable calculus problem involving partial derivatives along a surface

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  8. abobik

    Find the partial diameter error of the surface area of cylinder

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  9. M

    Find Isobaric Expansion & Pressure-Volume Coefficient for Solid

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  10. Silvia2023

    For this Partial Derivative -- Why are different results obtained?

    Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
  11. P

    What does this expression involving Partial Derivatives mean?

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  12. BloonAinte

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  13. N

    Correct Usage of Partial Derivative Symbols in PDEs

    Some may say that ##\frac{ \partial g }{ \partial t }## is correct because it is a term in a partial differential equation, but since ##g## is a one variable function with ##t## only, I think ##\frac{ dg }{ dt }## is correct according to the original usage of the derivative and partial...
  14. L

    B Question about the definition of a partial derivative

    I just started to study thermodynamics and very often I see formulas like this: $$ \left( \frac {\partial V} {\partial T} \right)_P $$ explanation of this formula is something similar to: partial derivative of ##V## with respect to ##T## while ##P## is constant. But as far as I remember...
  15. G

    I Understanding Covariant and Partial Derivatives in General Relativity

    In the 128 pages of 《A First Course in General Relativity - 2nd Edition》:"The covariant derivative differs from the partial derivative with respect to the coordinates only because the basis vectors change."Could someone give me some examples?I don't quite understand it.Tanks!
  16. J

    Calculating the partial derivative in polar coordinates

    Hello, I am trying to solve the following problem: If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...
  17. Delta2

    I From a proof on directional derivatives

    Given that the partial derivatives of a function ##f(x,y)## exist and are continuous, how can we prove that the following limit $$\lim_{h\to 0}\frac{f(x+hv_x,y+hv_y)-f(x,y+hv_y)}{h}=v_x\frac{\partial f}{\partial x}(x,y)$$ I can understand why the factor ##v_x## (which is viewed as a constant )...
  18. J

    I Partial Derivative of Convolution

    Hello, I am trying to calculate the partial derivative of a convolution. This is the expression: ##\frac{\partial}{\partial r}(x(t) * y(t, r))## Only y in the convolution depends on r. I know this identity below for taking the derivative of a convolution with both of the functions only...
  19. A

    Calculating specific heat capacity from entropy

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  20. L

    A Heisenberg equation of motion -- Partial derivative question

    Heisenberg equation of motion for operators are given by i\hbar\frac{d\hat{A}}{dt}=i\hbar\frac{\partial \hat{A}}{\partial t}+[\hat{A},\hat{H}]. Almost always ##\frac{\partial \hat{A}}{\partial t}=0##. When that is not the case?
  21. S

    A Index notation and partial derivative

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  22. D

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  23. J

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  24. F

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  25. P

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  26. R

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  27. M

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    Can anyone please help me to write partial derivative of Vxx w.r.t. r in terms of Vxx as shown in the hand written box at the end.
  28. LCSphysicist

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  29. S

    B Reconciling basis vector operators with partial derivative operators

    Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...
  30. BvU

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  31. R

    Can we take the partial derivatives of φ and ψ here?

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  32. A

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    If we have two functions C(y(t), r(t)) and I(y(t), r(t)) can we write $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?
  33. A

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  34. SchroedingersLion

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  35. Saptarshi Sarkar

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  36. currently

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  37. WhiteWolf98

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  38. George Keeling

    I Question about a partial derivative

    I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error: I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative \begin{align}...
  39. C

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  40. F

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  41. A

    I Partial Derivative: Correct Formulation?

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  42. A

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  43. Celso

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  44. Boltzman Oscillation

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  45. CCMarie

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  46. K

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  47. D

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  48. Peter Alexander

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  49. Jamie_Pi

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  50. C

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