# 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).

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1. ### Partial Differentiation of this Equation in x and y

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...
2. ### Estimating maximum percentage error using partial differentiation

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3. ### I Partial differentiation and explicit functions

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
4. ### Find that this partial differentiation is equal to 0

$$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$ They wrote that above equation is equal to...
5. ### Partial Differentiation -- If w=x+y and s=(x^3)+xy+(y^3), find 𝝏w/𝝏s

𝝏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.
6. ### Critical points and partial differentiation

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...
7. ### MHB Mastering Partial Differentiation: A Comprehensive Guide

Can you please help me ? I have tried to do it many times but I end up getting the wrong answer. Thank you in advance.
8. ### I Questions about Partial Differentiation Operations

1) If we have two functions C(y, r) and I(y, r) can we write: ∂C/∂I×∂I/∂r=∂C/∂r ? Can we also write ∂I/∂C=1/(∂C/∂I) ?
9. ### A Partial Differentiation in Lagrange's Equations

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...

14. ### I Integrating Equation 1: Understanding the Answer

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...
15. ### Partial Differentiation Laplace Equation Question

Homework Statement Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions: $$\frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0$$ u(x,0) = cos(x)...
16. ### Insights Partial Differentiation Without Tears - Comments

andrewkirk submitted a new PF Insights post Partial Differentiation Without Tears Continue reading the Original PF Insights Post.
17. ### I Partial differentiation

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...
18. ### Implicit partial differentiation

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 +...
19. ### I Calculating semi-major axis and minimum mass of an exoplanet

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 ±...
20. ### B A simple differentiation and partial differentiation

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.
21. ### Partial differentiation

Hello Is what I have done calculated here correct? Please correct me if I have done something wrong. Thanks in advance.
22. ### Confused About the Chain Rule for Partial Differentiation

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...
23. ### Partial differentiation problem

Homework Statement Homework EquationsThe Attempt at a Solution 1. If z=x+sin(##x^2##y) + ln y find ##\frac{\partial ^2z}{\partial x^2}## and ##\frac{\partial ^2z}{\partial y^2}## 2. Second order partial differentiation. 3. ##\frac {\partial z}{\partial x}## = 1 + ##cos(x^2y)## . (2x) =...
24. ### Partial differentiation of integral

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...
25. ### Evaluating a derivative by partial differentiation proof

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...
26. ### Partial Differentiation -- y deleted or ignored?

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...
27. ### Partial differentiation

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...
28. ### Partial differentiation question rocket trajectory

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.
29. ### Partial Differentiation Question

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...
30. ### Multi-variable Calculus : Partial differentiation

Homework Statement 2. The attempt at a solution By chain rule, which simpifies to, After this I am struck.
31. ### Partial differentiation problem, multiple variables (chain rule?)

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...
32. ### Partial Differentiation: second partial derivative

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...
33. ### Partial differentiation and partial derivatives

Homework Statement If ##xs^2 + yt^2 = 1## (1) and ##x^2s + y^2t = xy - 4,## (2) find ##\frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t}## at ##(x,y,s,t) = (1,-3,2,-1)##. Homework Equations Pretty much those just listed...
34. ### MHB Partial differentiation of an integral

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 -...
35. ### Partial Differentiation Identity Problem

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36. ### Partial differentiation question?

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...
37. ### How to Apply Partial Differentiation to V=f(x²+y²)?

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...
38. ### MHB Trouble with Solving a Partial Differentiation Problem?

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!
39. ### Partial differentiation with 3 variables

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...
40. ### Verify partial differentiation.

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...
41. ### Simple Partial Differentiation problem

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42. ### Change of variable for 2nd partial differentiation and higher.

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...
43. ### Simple Partial Differentiation Question

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) =...
44. ### Partial differentiation

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?
45. ### Partial differentiation: prove this general result

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...
46. ### Partial differentiation - Constants

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...
47. ### Unusual partial differentiation equation

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...
48. ### Partial Differentiation with Indicial Notation (Ritz Method for FEM)

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...
49. ### Partial differentiation: thermodynamic relations

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...
50. ### What is the meaning of partial differentiation in physics?

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...