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.

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Finding a closed-form optimization for a least squares function on a vector with two references (involving music)

I'm trying to solve a problem which is conceptually about music but shouldn't really require any major music theory background. I'll paste the prompt below. A polyphonic musical passage has $N$ notes ...
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Calculate the third partial derivative

If $u = vt^{\frac{1}{3}}$ and $v=h(y), y=it^{\frac{1}{3}}\sin(iwx)$ I need to determine $ u_{xxx}$ and $u_t$ I have : \begin{align} u_x &= u_v \cdot v_y \cdot y_x\\ &= t^{\frac{1}{3}}v_y(-wt^{\...
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Analysis - Prove a function that maps the unit Euclidean ball to R with bounded partial derivatives is uniformly continuous

I am stuck with this problem from my textbook and I cannot see the solution. I'm certain the solution is fairly simple and I am just missing the mark somehow. Any help would be appreciated. Suppose ...
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How to properly write down $\frac{\partial}{\partial xy } f(xy, x^2+3y-2)$

I am wanting to take the derivative with respect to $x$ of $f(xy, x^2+3y-2)$, so, I use the chain rule to find $$ \frac{\partial}{\partial x} f(xy, x^2-2) = y \frac{\partial}{\partial xy}f(xy, x^2-2) +...
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How to calculate the partial derivative function at any general point?

screenshot with transcription below: Consider the function $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by $$ f(x, y)=\left(1-\cos \frac{x^{2}}{y}\right) \sqrt{x^{2}+y^{2}} $$ for $y \neq 0$ ...
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Functions in two variables satisfying $f_{xy} f = f_x f_y$

What is the family of analytic functions $f : \mathbb{R}^2 \to \mathbb{R}$ which satisfy the condition$f_{xy} f = f_{x} f_{y}$? Is there any interesting significance to this condition? This holds for ...
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Isolated Point Singularities on Curves

I know that a curve $C: f(x, y) = 0$ is singular provided there is a point $(x_0, y_0) \in C$ for which $$\dfrac{\partial f}{\partial x}(x_0, y_0)=\dfrac{\partial f}{\partial y}(x_0, y_0)=0.$$ However,...
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Find the Partial Derivatives of $u= vt^{\frac{1}{3}}$ where $v=h(y)$ and $y=xt^{-\frac{1}{3}}$

If we have the following: \begin{align} y &= xt^{-\frac{1}{3}}\\ v &=h(y)\\ u &= vt^{\frac{1}{3}} \end{align} I need to determine $u_t$, $u_x$ and $u_{xxx}$ in terms of $\frac{dv}{dy}$ ...
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Why is the integral of the derivative of f over f equal to zero over closed curves? [duplicate]

I'm trying to show that if $f(z)$ is analytic an $|f(z) - 1| < 1$, then $\int_\gamma \frac{f'(z)}{f(z)}dz = 0$ over all closed curves $\gamma$. Presumably, I need to show that this is an exact ...
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Why is the derivative of f equal to the sum of its partials along its components?

I was trying to understand the development of the solution in this answer, where $\overline{f(z)}f'(z)dz = (u - iv)(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y})(dx + idy)$. The ...
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Full derivative of a scalar function wrt vector - chain rule

This confusion arose from this matrix calculus helper, the chart of total derivatives on page 23. The case with the scalar function of a vector of $n$ variables: it can be written as a product of a $1\...
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Partial derivative with two different limits

$$z = f \left( u(t), \ v(t)\right) $$ In my lecture notes, it is said that $$ \lim_{h\rightarrow0}\frac{f\left(u\left(t+h\right){,}\ v\left(t+h\right)\right)-f\left(u\left(t\right){,}\ v\left(t+h\...
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Is it analytic ? $f(z)=z^2\bar{z}$

Please find the question and its solution. My answer gone wrong dont know why, I think the answer marked in the question is wrong. My try: I solved for CR equations and replaced x=y=0 and found CR ...
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Two different partial differential equations from one equation

Let $z=(x^2+a)(y^2+b)$. Obtain the partial differential equation by removing constants $a$ and $b$. Attempt 1: $\frac{\partial z}{\partial x}=2x(y^2+b), \quad \frac{\partial z}{\partial y}=2y(x^2+a)$ ...
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If $f_x(x,y)>0$, $f_{xx}(x,y)<0$, $f_y(x,y)>0$, $f_{yy}(x,y)<0$ can $f_{xy} $ change sign?

That is, suppose we have a continuous and (at least) twice differentiable function $f(x,y)$ which is increasing but concave in each of its individual arguments$^*$, Note that these derivatives are ...
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Properties of Partial Derivatives?

Do Partial Derivatives have the property, for some general set of variables x,y,z, $$\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}=\frac{\partial x}{\partial z}$$ like exact derivatives ...
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Direction of the maximum increase for a piecewise function

Let $$f(x,y) = \begin{cases} \frac{x^3+y^3}{x^2+y^2} & (x,y) \neq (0,0)\\ 0 & (x,y)=(0,0) \end{cases}.$$ What direction at (0,0) gives the most incease in $f$? Attempt: The desired ...
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derivate is based on addition, is there a muliplication analogon?

like $$ \operatorname{f^o}(x) = \lim_{h\to 1} \frac{f(x*h)}{f(x)} $$ $$ \operatorname{f}(x)=e^x $$ $$ \operatorname{f^o}(x) = \lim_{h \to 1} e^{x*h}/e^{x} = \lim_{h\to 1} e^{x*h-x}=e^0=1 $$ does it ...
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Prove whether or not $f(x,y) = \frac{x^2y}{x^2+y^2}$ is differentiable at $(0,0)$.

I have the following function: $f: \mathbb{R}^2 \to \mathbb{R}$ with $f(x,y) = \frac{x^2y}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$. Now I want to proof that $f$ is differentiable in $(0,0)$, ...
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Can you always treat variables as constants in partial derivatives/integrals?

Ok so imagine you are given an expression: $xy : x=y$, & you need to take the partial derivative w.r.t. $x$ but you don't simplify (i.e. substitute x=y) --> $d/dx(xy)=y$. Now you've gotten a ...
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Can $\exp(-\frac{1}{x^2+y^2})$ be written as $xf(x,y)+yg(x,y)$ form? [closed]

Consider a $C^{\infty}$-function $F(x,y)$ on $\mathbb{R}^2$ defined by $$ F(x,y)=\begin{cases} \exp(-\frac{1}{x^2+y^2}) &\mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y)=(0,0) \end{cases} . ...
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Calculating the normal to a hyper surface

The hyper surface in my lectures is defined as a single function of coordinates which is a constant- $f(x)= c$ and it’s stated that the normal to the hyper surface is given as $x_\mu= \nabla_\mu f(x)$ ...
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38 views

Derivative of $h(t,x)=x\cdot x=x_1^2+\cdots+x_n^2$

Suppose that $x'=A(t)x=-A(t)^{T}x$, where $A(t)$ is a matricial function. What is the derivative of $h(t,x)=x\cdot x=x_1^2+\cdots+x_n^2$. I'm trying to prove that it is $0$, this way $h$ is a first ...
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Leibnitz integral rule on double integral

I am trying to compute the following partial derivative: $$ \frac{\partial}{\partial x}\iint_{\Omega(x)} f(x,\mathbf{w})d\mathbf{w} $$ with $f:\mathbb{R}\times \mathbb{R}^2\to \mathbb{R}$. In this ...
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1answer
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Calculating the partial derivative with respect to a matrix

Say, $y = W^T b$ where, $ W^T \in R^{2 \times 2} $ and $ b \in R^{2 \times 1} $. Now, we want to calculate: $\frac{\partial y}{\partial W}$. First, if we look at the dimensions: $y \in R^{2 \times 1}$....
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Using chain rule to prove the Implicit function theorem

In Analysis 2 by Terence Tao, the Implicit function theorem is stated as follow: Let $E$ be an open subset of $\mathbb{R^n}$. Let $f: E \to \mathbb{R}$ be continuously differentiable. Let $y = (y_1, ...
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Inconsistent derivatives with respect to a function when using the chain rule

I am trying to derive an expression for some partial derivatives in two different ways, but they seem to lead to inconsistent results. I have got the Cartesian components of a two-dimensional position ...
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1answer
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Partial derivatives and orthogonal relations

I am studying some stuffs relative to PDEs, and appear the following sentences that for are very artificial and I do not understand. By considering the relation on $x=a$ $$-A (\partial_x f - \...
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Confused about partial derivatives

I am having some issues understanding what should I keep constant and what not in certain cases when I take partial derivatives. Specifically in this kind of situation: say we have a function $$f(x,y) ...
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Why in tensors is the identity $\int{A_{\nu}\partial_{\mu}}\partial^{\nu}A^{\mu} = \int A_{\mu}\partial^{\mu}\partial^{\nu}A_{\nu}$ valid

I am trying to understand the and there is a step in the Derivation of something and there is a step which I cannot understand. Its basically a lack of understanding of tensors so any help would be ...
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When can we interchange operations involving partial derivative and integral, and how does Hamiltonian formulation affect that

Say we have an integral $$ B=\int dt \frac{\partial}{\partial q} f(t,q,\dot{q}), $$ where $t$ is an independent parameter, $q=q(t)$ is a dependent variable, $\dot{q}=dq/dt$ and $f$ is an arbitrary ...
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How to prove that a partial derivative of a combined function exists?

For one of my homework assignments the following function is given: $h: \mathbb{R}^3 \to \mathbb{R}$ with the following properties: The domain of $h$ is $\mathbb{R}^3$ All of its second order partial ...
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Derivative of Multivariate Function

Assume $M_\theta = \sum \theta_iM_i, \phi(M_\theta) \text{ is a differentiable function }\mathbb{R}^{n\times m} \rightarrow \mathbb{R}, \theta = [\theta_1 \dots \theta_N]$. I want to find $\frac{\...
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Don't linear combinations of partial derivatives entirely determine Jacobian?

I'm learning multivariable calculus for the first time, having just taken a course on theoretical linear algebra and group theory, and the instructor motivated the Jacobian definition of a function ...
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Differentiate a vector-valued function means to differentiate each of the components separately

If we have $f: \mathbb{R^n} \to \mathbb{R^m} $. We can write $f$ in terms of its components $f = (f_1, f_2, ..., f_m)$ We want to calculate the $partial \, derivative$ with respect to the $j^{th}$ ...
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Rigorous definition of derivative of a real-valued function

I am learning Real Analysis through the book Analysis 2 of Terence Tao and there is a notation of Partial derivative (in Definition 6.3.7 below). In the definition, the author uses the notation $\quad\...
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Derivative respect to a compose function

Given $x=r \circ \varphi$, why is it holds that: $$\frac{\partial f}{\partial x}= \frac{\partial (f \circ \varphi^ {-1})}{\partial r}$$ I really don’t get the rule that we use when the composed ...
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Question involving relation between directional derivative and the gradient of a function

Prove or disprove the following statement: Suppose a function has all its directional derivatives $$D_\hat{p}f$$ exist at a point $$(a,b)$$, then $$D_\hat{p}f(a, b) = (\nabla f)(a,b)\cdot \hat{p}$$ I ...
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Show that $u = 0$ in $B_1 (0)$ if we only assume that $u \in C^1 (B_1 (0))$.

Let $n \geq 2$ and $B_1 (0)$ be the open unit ball in $\Bbb R^n$. Suppose $u \in C^1 (B_1(0))\cap C(\overline{B_1 (0)})$ is a solution of linear PDE $$x \cdot Du=-u; x \in B_1 (0)$$ (a) Show that $u =...
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1answer
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Taking derivatives of a vector

I stumbled across the following question in a machine learning textbook: Let $x \in \mathbb{R}^{n}$. Let $y = $sin$(Ax) \in \mathbb{R}^{m}$. What is $\frac{\partial y}{\partial x}?$ What is its ...
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55 views

Show that $f(x,y) = \frac{x^2y^2}{x^2+y^2}$ is (totally) differentiable

I want to prove that $f(x,y) = \frac{x^2y^2}{x^2+y^2} , (f(0,0) = 0)$ is (totally) differentiable at $ \begin{pmatrix} 0 \\ 0 \end{pmatrix}$. I want to use the criterion that a function which is ...
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1answer
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Confusion with the partial derivative of a natural logarithm [closed]

Why do we solve the following example using chain rule? I would assume that the answer would be $18/x^1$.
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local minimum in 0

Consider the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by $f(x,y)=(y-x^2)(y-3x^2)$ Prove that: If $u\in\mathbb{R^2}\setminus\left \{0\right\}$ an arbitrary vector and $g:\mathbb{R}\...
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25 views

How does second partial derivative test be interpreted? [duplicate]

I know that what the formula is used to determine a point whether it has maximum, minimum, or saddle points; but, i do not understand how it is formulated. $$ H = f_{xx}​(x_0​,y_0​)f_{yy}​(x_0​, y_0​) ...
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60 views

$n$-th Derivative $\frac{d^{n}}{d x^{n}} e^{-\sqrt{x} |\omega|}$ via Recursive Product Rule

Let $g(x) = x^{-\frac{1}{2}}$ and $f(x) = e^{-\sqrt{x} |\omega|}$. I am trying to find an expression for the $M$-th derivative of their product: \begin{align} \frac{d^M}{dx^M} \left[ f(x) g(x) \...
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Help understanding how to input/simplify a certain partial derivative in order to solve

The Problem I have the following information from an article. I am not seeking help with the engineering side of this, but am more looking for information on how to write out / solve a portion of a ...
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1answer
32 views

Converting vector field from cartesian to cylindrical and finding the curl

The question is this: (a) Find the curl of the vector field $ 𝐯= y \hat{x} +𝑥 \hat{𝑦} +𝑥𝑦 \hat{𝑧}$ in Cartesian coordinates. (b) Rewrite 𝐯 in cylindrical coordinates. (c) Find ∇×𝐯 explicitly ...
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2answers
21 views

Partial derivatives in a function with “linked” variables

Let $x,y,z$ be three variables, $\phi=\phi(x,y,z)$ be a function of these variables and $c_1,c_2$ be two constants. If we are given that $$x=c_1\left(\frac{y}{z}\right)^{1/3} \tag{1}$$ $$y=c_1^{-3}x^...
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How to find the partial derivatives of the following nested expression?

I want to find the partial derivatives of the expression for $v_3(\boldsymbol{u})$ with respect to $u_1$, $u_2$ and $u_3$ from the expressions below. Here $\Phi$ denotes the cumulative distribution ...

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