Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [partial-derivative]

For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

The partial derivative of a function of several variables is the derivative of the function with respect to one of the variables, with the others held constant. The partial derivative, like the ordinary derivative, describes the rate of change of a function in a particular direction.

If $f = f(a_1, a_2, \dots, a_n)$ is a function of $n$ variables, then the partial derivative of $f$ with respect to the variable $a_i$ can be written as a limit:

$$\frac{\partial f}{\partial a_i} = \lim_{h \to 0} \frac{f(a_1, \dots, a_i + h, \dots, a_n) - f(a_1, \dots, a_i, \dots, a_n)}{h}.$$

Alternatively, this quantity can be denoted as $f_{a_i}$.

If the function has continuous partial second derivatives, then:

$$\frac{\partial}{\partial x_i}\left(\frac{\partial}{\partial x_j}\right)=\frac{\partial}{\partial x_j}\left(\frac{\partial}{\partial x_i}\right)$$

a result known as Schwarz's Theorem.

6917 questions
1
vote
0 answers

Schwarz's theorem for third order partial derivatives

I am supposed to check the below equation: $$ \frac{\partial^3 f}{\partial x^2 \partial y} = \frac{\partial^3 f}{\partial x \partial y \partial x} = \frac{\partial^3 f}{\partial y \partial x^2} $$ where $$ f(x,y) = x\sin^2y $$ I understand I could…
Comfy Cat
  • 89
1
vote
0 answers

Why does the expansion of this partial derivate give the following result?

$$(\cos \phi \frac \partial {\partial \rho} - \frac {\sin \phi} \rho \frac \partial {\partial \phi})(\cos \phi \frac {\partial g} {\partial \rho} - \frac {\sin \phi} \rho \frac {\partial g} {\partial \phi}) $$ $$=\cos ^2 \phi \frac {\partial ^2 g}…
Victor Fizesan
  • 11
1
vote
2 answers

Derivate of a complex function

I found in a paper the folowing derivative: $$\frac{\partial f(X(\theta),\theta)}{\partial \theta}=\frac{\partial f(X(\theta),\theta)}{\partial X(\theta)}\frac{\partial X(\theta)}{\partial \theta}+\frac{\partial f(X(\theta),\theta)}{\partial…
user137684
  • 175
1
vote
0 answers

How to derive gradients manually for backpropagation

We have the following feedforward equations: $z_1 = W_1x + b_1$ $a_1 = f(z_1)$ $z_2 = W_2a_1 + b_2$ $a_2 = y^* = \mbox{softmax}(z_2)$ $L(y, y^*) = -\frac{1}{N}\sum_{n \in N} \sum_{i \in C} y_{n,i} \log{y^*_{n,i}}$ Now, I'm trying to compute the…
py1123
  • 21
1
vote
2 answers

Derivative of $f(t-kx)$

If I have a function $f(t-kx)$ where $k$ is a constant and $t$ and $x$ are variables, then I know by looking at it that $\frac{\partial f}{\partial t}=-(1/k)\frac{\partial f}{\partial x}$ since it's obvious that changing $x$ is equivalent to…
kotozna
  • 153
1
vote
1 answer

how to tackle this partial differentiation problem

show that if $$f(x, y, z)=0$$ then $$\left ( \partial x \over \partial y \right )_{z}\left ( \partial y \over \partial z \right )_{x}\left ( \partial z \over \partial x \right )_{y}=-1$$ I don't know how to tackle this problem although I've…
Abdullah O. Alfaqir
  • 113
  • 5
1
vote
1 answer

Euler's theorem for this function

If $u= \arcsin\left(\dfrac{x+y}{\sqrt x+\sqrt y}\right)$ then show that : $$x^2\frac{\partial^2 u}{\partial x^2}+ y^2\frac{\partial^2 u}{\partial y^2}+ 2xy\frac{\partial^2 u}{\partial x \partial y}=-\dfrac{\sin u\cos 2u}{4\cos^3 u}$$ I tried using…
Aladdin
  • 282
1
vote
1 answer

Function-by-function derivatives commonly used in the physics litterature

In statistical physics, one is often presented with function-by-function derivatives. For instance, consider the derivation of the fundamental relation of thermodynamics from statistical physics. Let the entropy be: $$ S=\ln Z+\beta \overline{E} +…
Anon21
  • 2,581
1
vote
0 answers

Partial derivatives of radial basis function?

Assume I have a $N$-dimensional variable $X \in \mathbb{R}^N$ and a radial basis function $K(X,X')\rightarrow\mathbb{R}$ centered on $X' \in \mathbb{R}^N$: $$K(X,X')=-\exp\left(\frac{\lVert{X-X'}\rVert^2}{2h^2}\right)$$ with $$X=\begin{bmatrix} …
J.Galt
  • 961
1
vote
0 answers

How to find derivative of this integral?

I need to find the partial derivative $\frac{\partial}{\partial t} \int_{0}^{\infty}e^{-a\tau}I(x,t-\tau)\ d\tau$. The answer is given by $I(x,t)-a\int_{0}^{\infty}e^{-a\tau}I(x,t-\tau)\ d\tau$. What rule should I use to get the above answer?
Fatimah
  • 21
1
vote
1 answer

If $w=f(x, y),x=r\cos\theta, y=r\sin\theta$, prove $\frac{dw}{d\theta}^2+\left(r \frac{dw}{dr}\right)^2=\frac{dw}{dx}^2+\frac{dw}{dy}^2$

If $w = f(x,y)$, $x = r\cos(\theta)$, and $y = r\sin(\theta)$, show that $$\left(\frac{\mathrm{d}w}{\mathrm{d}\theta}\right)^2 + \left(r\frac{\mathrm{d}w}{\mathrm{d}r}\right)^2 = \left(\frac{\mathrm{d}w}{\mathrm{d}x}\right)^2 +…
pdf1234
  • 45
1
vote
1 answer

$\frac {\partial} {\partial y'}\bigg(\frac {dF}{dx}\bigg)$

In a part about the Euler - Langrange equation it is said that for a function $F(x,y,y')$ $$ \frac {dF}{dx} = \frac {\partial F}{\partial x}+ \frac {\partial F}{\partial y}y'+ \frac{\partial F}{\partial y'}y''$$ Now we wish to apply this to: $$\frac…
Joe Goldiamond
  • 1,094
1
vote
1 answer

Partial Derivative of ($x = $PX)

I'm trying to compute the partial derivative for ($x = $PX) where ($P$) is a Projection Matrix and $X$ is the World Space coordinates and $x$ is the image space coordinates. We have $$ {p_{3,4} = \begin{pmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\…
Y. A.
  • 13
1
vote
1 answer

Name for problem of reconstructing a function from partial derivatives?

Reconstructing a single variable function from its derivative is simply called integrating. But say we have an unknown multivariable function $f$ and know its partial derivatives. To illustrate for two variable $f$, Is there a name for the problem…
user56834
  • 12,925
1
vote
2 answers

Elementary Second partial derivative

If $V=\frac{xy} {(x^2+y^2)^2}$ and $x=r\cos\theta$, $y=r\sin\theta$ show that $\frac{\partial^2V} {\partial{r^2}}+\frac{1}{r}\frac{\partial{V}} {\partial{r}}+\frac{1} {r^2} \frac{\partial^2{V}}{\partial{\theta^2}}=0$ My attempt : $\frac{\partial…
Orestes Dante
  • 549
1 2 3
…
28 29