# Partial differentiation Definition and 9 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. ### Estimating maximum percentage error using partial differentiation

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...
2. ### 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...
3. ### 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...

6. ### I ∂^2u/∂x∂y = 0

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...
7. ### Insights Partial Differentiation Without Tears - Comments

andrewkirk submitted a new PF Insights post Partial Differentiation Without Tears Continue reading the Original PF Insights Post.
8. ### 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 ±...
9. ### 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.