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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Partial derivative of a function their variables depend on each other

if $z=F\left(x,y\right)$ and $y$=$\phi \left(x\right)$ Then is it correct to say that $z$ is just a function of a single variable which is $x$ ? and if we try to compute $\frac{\partial z}{\...
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1 vote
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Second partial derivative: What happens if $f_{xy}^2$ is larger than $f_{xx} f_{yy}$

Considering $(a,b)$ is a critical point of a funciton $f(x,y)$ and $D(x,y) = det(H(f(x,y))) =f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2}$ is the determinant of the hessian matrix from that function. If ...
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Finding a solution of partial derivative of a standard derivative

Considering the formula mentioned below, I arrived to the expansion as stated after performing a partial derivative with respect to the x co-ordinate: $$\nabla \left({\frac{dB}{dt} B}\right) = \frac{...
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Let in domain $G⊂R^2$, the $ f:G→R, f∈C^1(G)$, and $\frac{ ∂f}{∂y}(x,y)≡0$ in G. Is it possible to assert that $f$ does not depend on $G$?

Question : Let in the domain $G\subset \mathbb{R}^2$, the function $f:G\rightarrow R, f\in C^1(G)$, and $\frac{\partial f}{\partial y}(x,y)\equiv0$ in G. Is it possible to assert that the function $f$ ...
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Under what conditions can it be stated on G that f does not depend on y? [closed]

Task here. I know how to explain well about the independence of a function when G=R or G=R^2, but when G⊂R^2 I don't understand what to do at all, please help
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How do I go about this first and second order partial derivatives question? [closed]

Problem Let $f(x,y)$ be a function for which all its first and second order partial derivatives exist and are continuous and let ${\bf v}=\langle a,b\rangle$. Express $$(D_{\bf v}(D_{\bf v}f))(x,y),$$...
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2 answers
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Partial derivative of function that relies on given gradient of another function

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be differentiable such that $\nabla f(2,3)=(3,4)$ We'll define $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ by $g(x,y)=f(x^2-y+2 , y^3-x+3)$ And I'm asked to ...
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1 answer
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Partial and Total Derivative Double check

I haven’t done these in about 15 years, did a little googling and feel confident, but want to make sure First and second order partial derivatives (including cross partials) If $F(x,y)=3x^2+3xy-4x^{-...
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Mistake in proof of "double derivative test" in calculus textbook

I'm currently studying for a semester test in advanced calculus, and one of the topics covered is finding the local minima and maxima of a 3 dimensional surface. The first theorem that was proved was ...
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Order of the cross-product preference $T_u \times T_v$ vs. $T_v \times T_u$

To explain this question better, I was working through my lecture's problem sets and this problem came up: Vector Calculus 6th Edition, Anthony Tromba, Jerrold E. Marsden Consider the closed surface $...
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1 vote
1 answer
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Why $w(0,t)=w(L,t)=0\Longrightarrow w_t(0,t)=w_t(L,t)=0$

Let $w$ be a $C^2$ function in two variables, $x$ and $t$. The domain of $x$ is $[0,L]$ whilst the domain of $t$ is $t\geq 0$. Suppose that $w(0,t)=w(L,t)=0$. The apparently $w_t(0,t)=w_t(L,t)=0$. I ...
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4 votes
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Partial derivative of a recursive function

There are two functions:$$f(x,y):\mathbb{R}^2\to \mathbb{R}$$and$$g(z):\mathbb{R}\to \mathbb{R}.$$Both are differentiable. I have a function $G(x)=g(f(x,G(x)))$. I want to take the derivative with ...
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are the partial derivatives at $(0, \pi)$ defined? [closed]

Are the partial derivatives defined at point $(0, \pi)$? If so, what are their values? I calculated the derivative by definition and sin (x/y) received a zero value for y. am I correct?
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-6 votes
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Describe where the following function satisfy the Cauchy-Riemann equations, and where the function is differentiable. f(z) = z|z| [closed]

enter image description here I wanna know the solution of it. Thank you.
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Let $R = [a, x] \times [b, y]$, and let $f : R\to\Bbb R$ be a function of class $C^1$ Prove that $F$ is of class $C^2$ in $R$ [closed]

Let $R = [a, x]\times [b, y]$, and let $f :R\to\Bbb R$ be a function of class $C^{1}$. If $F : R → R$ is given by $F(x,y)=\int \int_{R} f(x,y)dA$ Prove that $F$ is of class $C^{2}$ in $\operatorname{...
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time derivative of work energy theorem

so I stumbled upon a step in my textbook which I can't do by myself, hopefully someone can help me :). It goes as follows $${\frac{d \bf{p}}{dt} \cdot \bf{u}} = \frac{d}{dt}(\frac{m \bf{u}}{\sqrt{1-\...
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Calculating the transition density function by finding the partial derivative of a conditional probability. [closed]

I've been given a process $Y_t=(1+t)B_t^2,t\ge0$ where $B_t$ is a standard Brownian motion and asked to find the transition density function $f(y,t|x,s)$. I've been instructed that $f(y,t|x,s)$ can be ...
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If $V = f(S)$ and $S = g(S_1, S_2, S_3)$ then what is $\frac{\partial^2V}{\partial S_1^2}$ in terms of $\frac{\partial^2V}{\partial S^2}$?

Assume $V = f(S)$ and $S = \alpha_1 S_1 + \alpha_2 S_2 + \alpha_3 S_3$, with $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{R}$. How can we express $\frac{\partial^2V}{\partial S_1^2}$ in terms of $\frac{\...
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1 answer
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Difference between heat capacity at constant pressure and volume

Define $C_V=T (\partial{S}/\partial{T})_V$ and $C_P=T (\partial{S}/\partial{T})_P$. Prove that $C_P-C_V=T(\partial{V}/\partial{T})_P (\partial{P}/\partial{T})_V.$ This is equivalent to proving $(\...
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What are the conditions for a function to be a unique implicit function?

My question is pertaining to part b. What are the conditions for a function to be a unique implicit function? Do we only have to check if the partial derivative at (x0, y0) evaluate to a number other ...
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Question about write differential as sum of partial derivatives

I learned that if i have a function $f:R^p\to R^q$ i can write differential of function like $df(a)(u)=\frac{\partial f}{\partial x_1}(a)u_1+\frac{\partial f}{\partial x_2}(a)u_2+...+\frac{\partial f}{...
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0 votes
1 answer
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Partial derivative of a composite function, given the definition of one of the functions [closed]

I need to find $\frac{∂f}{∂x}$ of $f(x) = \frac{1}{2}\phi(wx+b)$ where $\phi(x)=x^2$ I'm looking at something like $\frac{∂f}{∂x} = \frac{∂f}{∂\phi}\frac{∂\phi}{∂x}$, but I'm not sure how to do it ...
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-2 votes
1 answer
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How can we expand $(f(x)*d/dx)^n$? And why? [closed]

Which one is the correct expansion of $(f(x)d/dx)^2$? $(f(x))^2 (d^2/dx^2)$ or $(f(x)d/dx)(f(x)d/dx)$? And why?
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Converence of the derivative of the convolution

Let $z \colon \mathbb R^2 \to \mathbb{R}$ be $\mathcal C_B^{1,1}(\mathbb R^2)$. Fix $\epsilon>0$, choose $\rho \in \mathcal C^\infty $ a mollifier and consider the convolution of $z$: $$z_\epsilon(...
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2 votes
2 answers
203 views

Partial derivative, show that problem. L.H.S to R.H.S

This is a question from Advanced Calculus by David Wider. If $u=f(x,y),x=r\cos(\theta)$ and $y=r\sin(\theta)$ show that $$\frac{\partial u}{\partial x}^2+ \frac{\partial u}{\partial y}^2 = \frac{\...
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1 answer
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Partial derivatives in scalar field taylor expansion

$\newcommand{\v}[1]{\mathbf{#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\dd}[1]{\mathrm{d}#1}$ In our lecture notes we derived the following formula for the Taylor expansion of a scalar ...
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1 vote
1 answer
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Consistency checking the relation between total differentials using the Jacobian matrix

I have the following relationship between variables: $$\varepsilon = \Psi -\frac{v^2}{2}\tag{1}$$ A while back, I had an integral with respect to $v$ and I wanted to convert it to an integral with ...
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2 votes
1 answer
35 views

Norm of vector in directional derivate

Reading about the derivatives according to a direction, I found a definition saying that the norm of the vector v in the formula $\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} ...
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Do $\frac{\partial{z}}{\partial{\phi}}=0$ implies $\frac{\partial{\phi}}{\partial{z}}=0$? [duplicate]

If $\frac{dy}{dx}=0$, then $\frac{dx}{dy}$ is unbounded. I'v tried to derive $\nabla$ in sc and there is: $$ z=rcos(\theta)\\ \frac{\partial}{\partial{z}}=\frac{\partial{r}}{\partial{z}}\frac{\partial}...
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1 answer
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derivative of "length element"

I am trying to understand the derivation of this webpage about Liouville’s Theorem. Let us have a very small volume $V=\delta x\delta p$ in phase $(x,p)$ space. The total time derivative of $V$ ...
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-1 votes
1 answer
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Local invertibility of

Please check my understanding. (a) Let $f:\mathbb{R^2}\to\mathbb{R^2}$ defined by $F(x,y)=(x^2-y^2, 2xy)$. Calculate derivative matrix of $F$ and show $F$ is locally invertible except possibly at the ...
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6 votes
3 answers
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When is $d^{2} x$ =0?

We want to find $d^{2} z$ : $$ (4 x-3 z-16) d x+(8 y-24) d y+(6 z-3 x+27) d z =0 $$ So, in my book, we apply the differential operator d to the above equation : We use the product rule ; $$ (4 d x-3 d ...
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0 votes
1 answer
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Total derivation with respect to time.

Good evening everyone, I am trying to get a good understanding of total differentiation versus time. The problem I can't understand is the following. Starting with the basic national income accounting ...
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0 votes
1 answer
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Question Regarding Second Order Derivative Test for 2 Variables Function

I'm a new student to Calculus 2, and I'm currently having a hard time comprehending the meaning of the Second Order Derivative Test. The Second Order Derivative Test can be written as: $$D = f_{xx}(...
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partial derivatives of a lipschitz continous function with lipschitz continous gradient bound

Consider the function $ f: \mathbb{R}^n \rightarrow \mathbb{R}$ Does the following hold true if the function as well as its gradient is Lipschitz continuous? $ \big \| \partial_{x_i}f(\boldsymbol x^{k+...
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1 vote
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Deriving the partial chain rule using first principles

Can someone derive the chain rule for ∂/∂x(T) using first principle where T is a function of x(r,θ,φ),y(r,θ,φ),z(r,θ,φ) ? The equation goes like: ∂T/∂x=∂T/∂r(∂r/∂x)+∂T/∂θ(∂θ/∂x)+∂T/∂φ(∂φ/∂x) ...
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1 answer
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How many times is the composition of multivariable functions partially differentiable?

Let $m, n, p \in \{1,2,\dots\}$, let $q, r \in \{0,1,\dots\}$, let $D$ be a non-empty, open subset of $\mathbb{R}^m$, let $E$ be an open subset of $\mathbb{R}^n$, let $f : D\rightarrow E$ be $q$ times ...
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-1 votes
2 answers
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How to find the derivative of a function in the direction from point to point?

How can I find the derivate of a function? This is the exercise: $z = x^3 - 3y^3$ at $M(3;1)$ in the direction from point $M$ to point $K(6;5)$.
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How to approach the gradient of ArgMin problem containing the Euclidean Distance

How should I take the partial derivative of a function defined by an optimization with respect to $q$, e.g. $$ \frac{\partial}{\partial q} \underset{p \in \mathbb{C}}{\arg \min } \sqrt{\left(q_{1}-p_{...
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2 votes
1 answer
109 views

Determining the sign of the directional derivative and the partial derivatives on a surface

This is the question: The solution says: a) The surface is given by $z=f(x,y)$ If we see in graph as we move towards $\vec{u}=<5,0>$, $z$ increases, thus $D_{\vec{u}}f(2,-4)$ is positive. I'm ...
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2 votes
2 answers
94 views

What's the difference between $\frac{\partial}{\partial x}$ and $\frac{d}{d x}$?

I wonder if there is a difference between: $$\frac{\partial f(x)}{\partial x}, \ \frac{\partial}{\partial x}f(x), \ \frac{d f(x)}{d x} \ \mathrm{and} \ \frac{d}{d x}f(x)$$ I saw all of them in a ...
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6 votes
1 answer
89 views

$\sum_{i,j=1}^nx_ix_j\frac{\partial^2f}{\partial x_i\partial x_j}=0$ and $\nabla f(0)=0$ implies constancy of $f$ in $B_1(0)$

Let $B_1(0)$ be the unit ball in $\mathbb R^n$ centered at the origin. Assume that the function $f\in C^2(B_1(0))$. Prove that $1)$If $f$ satisfies $$\sum_{i,j=1}^nx_ix_j\frac{\partial^2f}{\partial ...
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1 answer
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Partial derivative of an inner product [closed]

How do I get the partial derivative of: $$\frac{\partial }{\partial w} \langle Y, J - XDiag(w)\rangle$$
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1 vote
2 answers
37 views

Show that $\frac{\partial u}{\partial t} = a\frac{\partial u}{\partial x} + b\frac{\partial u}{\partial y} = 1$

If $u=f(r,s)$, $r=x+at$, $s=y+bt$ and $x$, $y$ and $t$ are independent variables and $a$ and $b$ are constants. Show that $\dfrac{\partial u}{\partial t} = a\dfrac{\partial u}{\partial x} + b\dfrac{\...
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-2 votes
0 answers
41 views

How to remove gradient from algebraic expression

So, I found the Schrödinger's Equation (shown below), and I found out I can expand it into two or three dimensions, as well, by simply adding another term $\frac{\partial^2\psi}{\partial z^2}$, for ...
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Derivative of this matrix expression?

Assume that I have the following expression: $$f(x,c) = \int_{0}^{x}g(\mathbf{\Psi}(x)^\intercal\mathbf{c}) dx$$ where $f(x,c)$ is a scalar, $x$ is a scalar, $\mathbf{\Psi}(x)$ is a $N$-by-$1$ vector, ...
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1 vote
1 answer
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When $f_x=f_y$, does that mean that all double partial derivative are the same?

While working with some partial derivatives, I noticed that some functions' partial derivatives are the same (e.g. $f_x=f_y$). For example, both partial derivatives of $e^{2x+2y}$ are $2e^{2x+2y}$. ...
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1 vote
2 answers
60 views

Find all of the extreme values using Lagrange Multipliers

I need to find all of the extreme values of the function $x^2+y^2+z^2$ constrained to $x^2+2y^2-z^2-1=0$ the problem is that I get this system that I have no idea how to solve: $2x=2\lambda x$ $2y=4\...
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0 answers
16 views

PDE with mass immigration and mass killing at zero

I am interested in the following PDE: Let $1> \alpha > \frac12$ and $$\frac{d}{dt} f(x,t) = \frac{d}{dx} f(x,t) + \alpha (x+1)^{-\alpha -1} \left[1- \exp\left(-f(0,t)\right)\right], \mbox{ for }...
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32 views

Interchange of differentiation and integration sign.

Let $f:\Omega\to \mathbb C$ be a function (where $\Omega$ is an open set) and suppose $\phi:\mathbb C\times \Omega\to \mathbb C$ be a function such that $f(z)=\int_C \phi(\zeta,z)d\zeta$ where $C$ is ...
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