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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6 views

How to solve Directional diretive indeterminate form

I have the function f(x,y) defined in this way: $$f(x,y)= \left\{ \begin{array}{c} {x^2 + y^2 +x\over x^2 + y ^ 2} & \text{if (x,y) $\ne$ (0,0)} \\ {1} &...
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Derivative of a function with two inputs

Suppose I have a $f(p,D)=pD$, now as I previously understood we will have to take partial derivative here and then form $2$ separate equations. But if my $p$ and $D$ variables are correlated, that is ...
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Vector function derivation

Let $h_k,v_k\in\mathbb{C}^{N\times1}$ and $C_k,D_k \in \mathbb{C}^{N\times N}$. $K_r$ and $\sigma_{k}^{2}$ are constant. $h_k^H$ denotes the conjugate transpose of $h_k$. A function is formulated as: ...
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15 views

Numerical $n$-th order mixed partial derivative

To start with, this might be a naive question since I do not have a lot of experience with numerical analysis. Let $f(\boldsymbol{x})$ be a function in $M$ variables, i.e. $f:\mathbb{R}^M\to\mathbb{R}...
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24 views

partial derivative of a function of two R.V.s respect to the expected value of one R.V.

Consider two independent random variables $X$ and $Y$, and $X \geq 0$, $Y \geq 0$. Let $Z = f(X, Y)$ Is it possible to calculate the partial derivative of $\mathbb{E}(Z)$ respect to $\mathbb{E}(X)$, ...
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37 views

commenting on whether $f'(x)$ is even or odd.

The question is as follows: Let $f(x)$ be a differentiable function $\forall x,y \in \Bbb R$ and $$f(x-y),f(x),f(y),f(x+y)$$ are in AP then comment whether $f'(x)$ is even or odd (given $f(0) \neq 0)...
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how to prove that this problem?

Is there partial laplace derivative equations? I am so confused. Show that the function provides the equation. \begin{equation} \label{simple_equation0} u = {\varphi }(xy)+\sqrt{xy}{\psi}(\frac{...
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1answer
37 views

Higher order derivatives and the chain rule

So here I have an assignment about higher order derivatives and the chain rule, and a relation to be proved: Show that for a rotation in the plane$$\begin{bmatrix}u\\v \end{bmatrix} =\begin{bmatrix}\...
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25 views

Lipschitz continuity and boundedness of derivatives.

Suppose $f(x)$ is a function that has derivative in the domain of our concern. We all know that if the derivative is bounded then $f$ is Lipschitz continuous. I was wondering if the above statement is ...
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31 views

Multivariable Calculus, more specifically flux integral

So the question gives vector field $F(x,y,z)=(x^2,y^2,z^2)$ and it is flowing out of a sphere of $(x-1)^2+(y+1)^2+(z-2)^2 \leq 4 $. Using Guass's rule, I transformed it into a triple integral while ...
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What are some strategies for computing derivatives in matrix calculus?

In single variable calculus, the chain, product, and power rules are very straightforward. In vector calculus, it's a bit more involved. I frequently find myself having to do a lot of work to arrive ...
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How to prove the result of an integral of partial derivatives

I have proof for question a but I cannot proof question b. How should I prove question b?
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Manipulating derivatives

I am starting to learn some multivariable calculus and had a number of questions on some of the simplifications of differential expressions as I have not found a good resource that guides through. 1)...
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41 views

How would I write the product of something whilst also omitting an element?

I have a function $$f(x_1,x_2,...,x_n) = \prod_{i=1}^{n}x_i^{\alpha_{i}}$$ I want to take the partial derivative $$\frac{\partial f}{\partial x_k}$$ Now I believe this will look like $${\alpha_kx^{\...
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25 views

Find local extrema of of the following multivariable function

A following function is given: $$ f(x,y,z)= x^2 + \frac{2}{x} + (2z+y)^2 + y^2 + \frac{2}{2z+y} + \frac{2}{y} $$ I know, that i have to start by calculating partial derivatives in respect to x, y and ...
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16 views

Exchanging derivative with evaluation

Suppose I have a continuously differentiable function of three variables, $\gamma(x, y, z)$, which satisfies $$ \gamma(x, y, x) = y. $$ Does it then follow that $$ \left[\frac{\partial \gamma(x, y, z)}...
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26 views

Acceleration $\mathbf a$ in function of the velocity $\mathbf v$

In the Physics.SE, I have seen this very nice question where the acceleration $\mathbf{a}$ is obtained using the differential operator $\boldsymbol \nabla$. $$\mathbf{a}=\frac{d\mathbf{v}}{dt}=\color{...
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17 views

Interchanging partial differentiation with evaluation along a curve

Suppose I have a continuously differentiable function of three variables, $\gamma(x, y, z)$. Is it always valid to interchange the order of the operations of evaluation at a point, and differentiation?...
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35 views

Take derivative of matrix

A part of an objective function is: $$F=\|H-\mu_H\|_F^2$$ And we have: $$\mu_H=\frac{\Sigma H}{n_H}$$ In fact, $\mu_H$ is the average of $H$ in one dimension and is repeated $n$ times in which all ...
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proving that $\frac{\partial t_k(L(\textbf{x}))}{\partial l_i}=\frac{\partial t_k(\textbf{x})}{\partial x_i}\Bigg|_{\textbf{x}=L(\textbf{x})}$

Imagine that I have the following functions: $T,L:\mathbb{R}^{n} \longrightarrow \mathbb{R}^n$, such that: $T(\textbf{x})=(t_1(\textbf{x}),...,t_n(\textbf{x}))$ and $L(\textbf{x})=(l_1(\textbf{x}),...,...
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How do I solve the following problem

Let's suppose I have the following function $f$: $$f: \mathbb{R}^2\setminus (0,0) \to \mathbb{R}$$ $$(x,y) \mapsto f(x,y) = \frac{1}{x} - \frac{1}{2y} - \frac{y}{2}$$ If I want to find the maxima ...
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43 views

Given $\phi$ a mapping. Prove that for each $\mathit{i}$, $\sum_{j=1}^n \partial_{x_j}(\mathbf{cof} \mathit{D} \phi)_{ji} \equiv 0$

Let $\phi \in \mathit{C}^2 (\mathbb{R}^n , \mathbb{R}^n)$. Let $\mathbf{cof} \mathit{D} \phi$ be the cofactor of $\mathit{D} \phi$ (the Jacobian matrix of $\phi$). i.e. $$(\mathbf{cof} \mathit{D} \phi)...
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Is the partial derivative of a constant always zero?

I'm trying to get my head round using the multivariable chain rule to find exact derivatives. For example I want to find the exact derivative(using partial derivatives) of, $$r^2=x^2+y^2$$ Where r is ...
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$g(x,y)=f(\sqrt {x^2+y^2})$ for (x,y) is not zero vector. Show that $y\frac{\partial g}{\partial x}=x\frac{\partial g}{\partial y}$.

Suppose $f: \mathbb{R} \to \mathbb{R}$ is differentiable. I think it need Euler's theorem to solve it. But I do not know the homogeneous degree of g. My attempt is $$g(tx,ty)=f(\sqrt {(tx)^2+(ty)^2})=...
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33 views

Show that $x\frac{\partial g}{\partial x}+y\frac{\partial g}{\partial y}=0$ when g$\begin{pmatrix}x \\ y\end{pmatrix}$ = $f({\frac{x}{y}})$

Can I compute $\frac{\partial g}{\partial x}$ to be $\frac{df}{dx}$? The reason is I think $\mathit{f}$ is a one variable function. So $$\frac{\partial g}{\partial x}=\frac{df}{dx}=\frac{1}{y}f'$$ ...
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20 views

Partial derivative with respect to U

I have the following equation dealing with matrices X, U, S. The lambda is scalar. $e = \arg\min_U |XU - S|^2 + \lambda |U|^2$ I have to solve derivative of this with respect to U and set it equal ...
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Partial differentiation with 4 variables $x, y, z, u_{(x, y, z)}$.

Let $F = F_{(x, y, z, u)}$ and $u = u_{(x, y, z)}$ be a differential function of $x, y, z$ implicitly defined by $f_{(x, y, z, u)} = 0$. Find the expressions for $\frac{\partial u}{\partial x}$, $\...
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1answer
19 views

objective function derivative

Let's assume I have an objective function: $f(x) = x^2-b$ If I find partial derivative w.r.t $b$, I get: $\frac{\partial f }{\partial b} = -1$. But if I want to minimize it, I would set it to zero:...
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22 views

If $V$ does not depend on $t$, how to show $V_{t} = V_{t+s}$, for $s\in \mathbb{R}$

I am reading a paper, and it says the following: Let $V = V(x(t))$, i.e., a function of state $x(t)$ but not depend explicitly on $t$. Now consider a mapping $$\tilde{t} = t+s, $$for any $s\in \...
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Partial derivative with respect to X when X is a subscript?

How do you take the derivative of a variable when that variable appears as the subset of another variable? i.e. Would $\frac{\partial}{\partial i_1}[a_{i_1j}x_{i_1}]$ equate to $a_{i_1j}x_{i_1}$ ...
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Differentiable norm $p$, show that: $\forall x\in V$ $\exists Y\in$ Lin$(V,\mathbb{R})$, so that $|Y(z)|\leq p(z)$ $\forall z\in V$ and $Y(x)=p(x)$

This is the problem which I'm struggling with: "Let $V$ be a finite vector space, let $p$ be a norm on $V$, with $p$ being differentiable on $V$\ $\{0\}$. Show that: For every $x\in V$ exists a $Y\...
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What is the mathematical significance of the integral of the normal derivative?

I found from Advanced Engineering Mathematics by Kreyszig that for the region $R$ and a scalar function $w(x, y)$ the following holds: $$\iint_R \nabla^2 w \ dxdy = \oint _{\partial R} \frac{\...
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geometric meaning of $\partial_{xy}f(x,y)?$ [duplicate]

I know that $\partial_x f(x,y)$ gives us the slope of the tangent line at the point (x,y) in the x direction. And $\partial_{xx}f(x,y)$ gives us the rate of change of that same slope. The same happens ...
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Find the value of the partial derivative.

Find the value of the derivative $$\left(\dfrac{\partial}{\partial x}\right)^m\left(\dfrac{\partial}{\partial y}\right)^n\left\{\dfrac{1}{z[a^2+(b+z)^2]}\right\}~=~?$$where $~z^2=r+sx^2+ty^2~$ and $~r,...
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30 views

Partial Derivative of Mean Absolute Error

I have a Linear Function, $Y= a+bX$. The Mean Absolute Error would be $$f(a,b)=\frac1n \sum |y-(a+bx)|$$ To find the partial derivative I used this formulae $$\frac{\partial f(x,y)}{\partial x} = \...
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43 views

Derivative of Jacobi elliptic function

Let $k \in (0,1)$ and $w>0$. Consider the function $\varphi: \mathbb{R} \longrightarrow \mathbb{R}$ given by $$\varphi(\xi)= \frac{\sqrt{2}k}{\sqrt{k^2+1}}\cdot \text{sn} \left(\frac{\xi}{\sqrt{w}\...
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42 views

Truncation error in finite difference approximation of mixed derivative

In a textbook (https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781119083405.app1) I came across a way of deriving a finite-difference discretization of the mixed derivative $\frac{\partial^2 f}{\...
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Volume of the rectangular box with open top by enquality of geometric and hormonic means

A rectanglular box without lid made of 12 m^2 of cardboard. Find the volume by appropriate usage of inequalities of geometric and geometric means
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Language and Algorithm to Take Derivatives in Function Notation

I want to algorithmically find derivatives and partial derivatives of expressions written in function notation. For example, I can type "d/dx f(x, y(x), g(x) * h(x))" directly into Wolfram Alpha, ...
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1answer
29 views

Integrating unit vectors

If I want to find a function $V(r)$ such that ${\bf F} = -\nabla V$, where ${\bf F} = f(r)\hat{\bf r}$, how do I integrate ${\bf F}$ along the radial direction?
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11 views

Simplify partial derivative expression

I have the following partial derivative that I am not sure how to completely simplify: $$\begin{eqnarray} \frac{\partial}{\partial c} (\frac{\partial c}{\partial x_i} \frac{\partial c}{\partial x_i}) \...
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1answer
31 views

Is this a particular case of the Stoke's Theorem? How to prove the equality?

I'm doing my Calculus III homework and I'm stuck in a question. It seems to be a particular case of the Stoke's Theorem but I'm not sure. The problem is: Be $B$ a triangle with vertex $(0,0)$, $(1,0)$...
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22 views

Mixed Total Derivative

Is there a mixed total derivative? I'm think of something like $\frac{d^2}{dydx}F(x,\ y(x)) = \frac{d}{dy}(invert\_perspective(\frac{d}{dx}F(x,\ y(x))))$. So if $F = 3xy + 5x + 7y$, where $y = 4x$, ...
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37 views

Compute Gradient and Hessian of following function

can anyone help me and give me directions to compute the gradient and the hessian matrix of this function : $$f:\mathbb{R^n}\to\mathbb{R^+}$$ $$f(x)=\|Ax-(x^TAx)x\|.$$ Thank you
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115 views

Second-Order In(exact) ODEs

The second total derivative of $F(x,\ y(x))$ is $F_y y'' + F_{yy}(y')^2 + 2F_{xy}y' + F_{xx}$. Thus by analogy to first-order exact ODEs, if one notices a second-order ODE where this pattern equals ...
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Unrecognised Partial Derivative Notation

I'm trying to brush up on my calculus for my job search and I'm really struggling to understand something basic that is frustrating me. The question is given by: If w $\in \mathbb{R}^d$ and x $\in \...
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20 views

$\partial_1f(x,t)=\partial_2f(x,t)$ and $f(x,0)>0 \implies f(x,t)>0$

I would be very glad if somebody could give me some advice on how to solve this problem: "Let $f\in C^1(\mathbb{R}^2,\mathbb{R})$ with $\partial_1f(x,t)=\partial_2 f(x,t)$ $\forall (x,t)\in\mathbb{R}^...
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21 views

Implication of equal partial derivatives

I am given the following problem: Let $f \in C^1 (\mathbb{R^2};\mathbb{R})$ and $\partial_1f(x,t) = \partial_2f(x,t)$ for every $(x,t)\in \mathbb{R^2}$ and $f(x,0) > 0 $ for every $x \in \mathbb{...
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1answer
27 views

Partial derivative reciprocal theorem

Is the reciprocal of a partial derivative always equal to 1 divided by that partial derivative? I.e. Is it true that; $$\frac{1}{\frac{\partial x_1}{\partial x_2}} = \frac{\partial x_2}{\partial x_1} ...
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56 views

Partial differentiation - chain rule

Assume I have a function f(x,y) and I transform the variables using the following: \begin{equation} m = 2x + y \end{equation} \begin{equation} n = x - y \end{equation} Or any other similar linear ...

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