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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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Factorising functions out of partial derivatives

I have been doing that work that requires me to use the chain rule on second order partial derivatives to replace variables (x, y) with (u, v) where u and v are functions of x and y. My question is ...
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1answer
28 views

Directional Derivative problem on GRE practice exam

The question reads: Let $g$ be the function defined by $g(x,y,z) = 3x^2 y + z$. What is the best approximation of the directional derivative of $g$ at the point $(0,0,\pi)$ in the direction of the ...
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0answers
14 views

hessian calculation with division by zero for first derivative

I am wanting to calculate the hessian from an example dataset that keeps evolving. At one particular data set, one of the parameters (z) does not change, which causes a division by zero for the ...
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1answer
21 views

Derivation Numerical Method with partial derivatives, vectors, matrices and scalar product

I need help in finding a way to combine the equations \begin{equation} \frac{\partial J}{\partial W} \cdot \delta W = \langle Y_M^T (\eta^{'}(Y_M W) \odot (\eta(Y_MW)-C)),\delta W \rangle \end{...
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1answer
15 views

Problem in understanding Chain rule for partial derivatives

I'm having trouble understanding the chain rule for partial derivatives. If I'm given that $\omega=f(x,y)$ where $x$ and $y$ are functions of both $t$ and $r$, then by chain rule I can write that: $$\...
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2answers
33 views

Find partial derivatives of $f(x)=\|x\|^\alpha$

Find partial derivatives of $f:\mathbb{R}^n\rightarrow\mathbb{R}$ $$f(x)=\|x\|^\alpha$$ outside of $(0,0)$ when $\alpha\in\mathbb{R}$. What values does $\alpha$ have to take for the partial ...
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1answer
41 views

A chain rule for the angle

My question is fairly basic, namely can I do the substitution below? $$ \frac{\mathrm{d}\theta(t)}{\mathrm{d}t} = -\frac{1}{\sin(\theta(t))}\frac{\mathrm{d}\cos(\theta(t))}{\mathrm{d}t} $$ If not, ...
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Critical points (Undefined partial derivatives) and KKT condition

I am going through the contents of KKT conditions. But it seems to deal with only cases that partial derivatives of the Lagrangian function $L$ being nonnegative. Is there any case where the local ...
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1answer
19 views

Derivative of multivariable piecewise function

I want to know how I can make the derivative of this piecewise function respect to the X variable. I know that in the point (0,0) you have to use the definition but I need the general derivative of ...
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0answers
19 views

How by using rotation matrix to relate polar $\frac{\partial}{\partial \rho}$ , $\frac{\partial}{\partial \phi}$ to Cartesian partial derivatives

How by using rotation matrix to relate the $\frac{\partial}{\partial \rho}$ and $\frac{\partial}{\partial \phi}$ to Cartesian partial derivatives? We do not want to use chain rule. The rotation ...
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derivative with respect to a function

I am a bit confused in evaluating the partial derivative: I have this equation $y(t)=x(t)+\dot{x}(t)\\$, where $\dot{x} = \frac{dx}{dt}$ $z=f(x,y,\dot{x})$ for example $z=\frac{(x-y)^2}{\dot{x}^2}$...
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How to calculate the partial derivative of first hidden layer in a neural network with 2 hidden layers

First off, this video explains how to retrieve the partial derivative of the neurons in the second hidden layer, but I am a little unsure if my calculations on how to retrieve the partial derivative ...
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1answer
24 views

Khan Academy, Example: Computing Partial Derivative

Struggling in following this problem and it's solution. Problem: $f(x, 2) = 8x^2$ Solution: $\frac{d}{dx}f(x, 2)=\frac{d}{dx}(8x^2)=16x$ $x=3$ $16(3)=48$ I got to the part where it's $8x^2$ - ...
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1answer
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How to find the respective terms on this problem on Partial Derivatives

I was studying about partial derivatives and I got confused by this problem. I'm asked to prove that $$(\frac{\partial \omega}{\partial \theta})^2+ \frac{1}{r^2}(\frac{\partial \omega}{\partial r})^2=...
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1answer
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why my solution to $\frac{\partial tr(ABA^{T})}{\partial A}$ is wrong?

Given $A$ and $B$ are matrix,I know the true answer of the derivative $\frac{\partial tr(ABA^{T})}{\partial A}=A(B+B^{T})$ However, I don't know why my solution is wrong? Here is my solution: " First ...
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Can we cancel differentials in partial derivatives? $\frac{\partial q}{\partial x} = \frac{\partial \dot{q}}{\partial \dot{x}}?$

I'm not sure whether this is better suited to the physics site, but I'm trying here first. My analytical mechanics professor, while discussing some generalized coordinate $q(x(t),y(t))$, used the ...
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1answer
59 views

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and $f$ is differentiable at z.

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and f is differentiable at $z$. The $\bullet$ denotes the dot(inner) product. $\nabla$ is the gradient. $...
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1answer
27 views

How to reduce a second-order differential equation given a substitution

Given the general partial differential equation $$a \frac{\partial^2u}{\partial x^2} + 2b \frac{\partial^2 u}{\partial x \partial y} + c\frac{\partial^2u}{\partial y^2} = 0,$$ if $ac - b^2 > 0$, ...
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discontinuities in first derivatives of PDEs

In the lecture notes (Oxford University, Applied Partial differential equations) it says that $u$ is continuous and only first derivatives of $u$ may be discontinuous across some curve $C$ in the $(x,...
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1answer
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Step in the proof of the adjoint representation of the Lie bracket

Let $a: \mathbb{R}^2 \rightarrow M$ a differential map such that $a(s, 0) = p$ for all $s \in \mathbb{R}$. Let $\gamma : \mathbb{R} \rightarrow T_p M$ the path given by $\gamma(s) : = \frac{\partial}{\...
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0answers
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Separation Of Variables Laplace Equation

Using the Separation of Variables technique, solve 2D Laplace’s equation $$\Delta u = 0 \quad for \quad (x,y) \in \ (0,a) \times (0,b)$$ with $a,b \in \mathbb{R}$ , subject to boundary conditions: $...
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Partial derivatives of a homogenous function

See demo of the theorem here : https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/hom/t) This is actually a follow up question on this one : Help to understand the proof of partial ...
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1answer
23 views

How to find partial derivative at a point? [on hold]

Consider the surface given by $$z=f(x,y)=\dfrac{-x^2}{2}-y^2+\dfrac{25}{\pi}$$ Find $z$ at $(1,2)$ So I know how to find the partial derivative of the function but what do I do with the points?
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prove laplacian equal zero [closed]

how I can find the first and second derevative respect to x?
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1answer
20 views

Vector function derivative

If a function $f(t,x)$ has $x \in \mathbb{R}^{2}$, what is the partial derivative $\frac{\partial{f}}{\partial{x}}$? Thank you greatly.
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2answers
87 views

Differentials in Multivariable Calculus

Does the idea of composing/decomposing the fraction notation of the derivative from/into differentials apply in multivariable calculus? I realize that this practice is considered non-standard and ...
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2answers
26 views

How can I determine whether or not a function exists given two partial derivatives?

"Can there exist a $C^2$ function $f(x,y)$ with $f_x = 2x-5y$ and $f_y=4x+y$"? Given this question, am I simply to take the second derivative of these functions to prove the equivalence of the mixed ...
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0answers
58 views

Calculate $\lim_{a \to 0}\frac{2^n-(2-a)(2-2^2a)(2-3^2a)…(2-n^2a)}{2^na}$

I have tried every way I know to calculate this end (divide $a^n$, divide $2^n$, derivation) but without any result. Can you help me calculate it? and thank you very much. $$ \lim_{a \to 0}\frac{2^n-...
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radical of the Jacobian ideal

Let $R=\mathbf{k}[X_1,\ldots,X_n]$. Definition: For $f\in R$, let $\mathcal{J}_f$ denotes the ideal generated by the partial derivatives of $f$ (Jacobian ideal), namely $$\mathcal{J}_f = \left\...
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1answer
20 views

Rewriting the two dimensional heat equation

Consider the heat equation $$k\nabla^2u=k\left ( \frac{\partial^2u }{\partial x^2}+ \frac{\partial^2u }{\partial y^2} \right )=\frac{\partial u}{\partial t}$$ in two dimensions, where $k$ is a ...
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1answer
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Finding the $2n+1$ th derivative of $\frac{y^{2n+1}xy}{1-x^2y^2}$ with respect to $x$.

$f(x,y) = \frac{y^{2n+1}xy}{1-x^2y^2}$. I made the following table: \begin{align} & 2n+1 = 1 \implies f^{(1)} = \frac{1!y^2(1+x^2y^2)}{(1-x^2y^2)^2}\\ & 2n+1 = 3 \implies f^{(3)} = \frac{3!y^...
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1answer
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differentiate $g(f(x),x)$ with respect to $f(x)$

Suppose I have a function $g(y,x)$ which is differentiable with respect to both arguments. I know that $y=f(x)$ is bijective, and thus the inverse function exists $x = f^{-1}(y)$. My question is when ...
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35 views

Finding the approximate error using partial differentiation

Good day! So, I need to solve for the approximate error in the common logarithm of the product of two numbers $x$ and $y$. I know the process, wherein I will have to find the total differential of ...
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1answer
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What is the derivative of $J(s)=\alpha { s }^{ T }(I-W)s+(1-\alpha ){ \left\| s-h \right\| }_{ F }^{ 2 }$ w.r.t. $s$?

$\alpha$ is a scalar, $W$ is a matrix, s and h are vectors, I is identity matrix. I know the derivative is as follows: $\frac { \partial J(s) }{ \partial s } =2(I-\alpha W)s+2(1-\alpha )h\quad$ but I ...
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$\nabla g(x) = (x^{T} x)^{m+1}$ derivation rule

I think my calculations of $\nabla g(x) = (x^{T} x)^{m+1}$ are wrong...mostly the last part. Can someone help me? Given $g(x) = (x^{T} x)^{m+1}$ we say that $f(x) = x^{T}x = \langle x,x \rangle = \...
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2answers
31 views

Difference between ordinary and partial differential equations [closed]

Being an undergraduate student I find difficult to understand the perfect differences between normal and partial differential equations. Elaborate the answer
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1answer
34 views

Connected Dirichlet and Neumann conditions

Im working on a project where we have to solve (numerically) the temperature distribution in an apartment, using Laplace equation. The following is a simplified model for that apartment, which ...
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21 views

Conjugate complex coordinates

This is the question and the problem is we didn't take multivariable calculus to understand these steps, where did it come from? When I search multivariable chain rule I get a different result than ...
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2answers
37 views

Which theorem says that $\displaystyle \int_{f_1(x)}^{f_2(x)}\frac{\partial M}{\partial y}\,dy=M(x,f(x_2))-M(x,f(x_1))$?

I want to explain where this equality comes from. I'm working with a proof of Green's theorem. Thanks very much.
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How to compute the partial derivative of the Hamiltonian?

The Hamiltonian associate to the Lagrangian $L$ is defined as $$H(p,x) = p\cdot v(p,x)-L(v(p,x),x).$$ We also make the following important hypothesis: suppose that for all $p,x\in \mathbb{R}^n$ the ...
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Question about the derivation of the charecteristics of the Hamilton Jacobi Equation.

I am reading Evan's book on PDE where he mentions the Hamilton Jacobi Equation as follows: $$G(Du,u_t,u,x,t) = u_t + H(Du,x) = 0$$ where $Du=D_xu = (u_{x_1},u_{x_2},...,u_{x_n}).$ Then writing $q=(p,...
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1answer
36 views

Partial Derivatives, find rate of change

I have done part a) of this question. I am confused about part b), as it doesn't say determine rate of change of temperature with respect to anything, so I am confused. Would it be ∂T/∂x + ∂T/dy ?
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1answer
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Can a divisor of derivation be cancelled out

Suppose I have this equation: $$ \frac{\partial N}{\partial x \partial t} = \frac{\partial^2\phi}{\partial x^2} $$ Can I cancell $\partial x$ and become: $$ \frac{\partial N}{\partial t} = \frac{\...
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1answer
41 views

Deriving partial chain rule using total derivative chain rule

Let $a,b : I \to \mathbb{R}$ be differentiable functions, and $m: I \to \mathbb{R}^2$ be a function such that $m(t)=(a(t),b(t))$. Further, let there be a differentiable function $f:\mathbb{R}^2\to\...
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0answers
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Find the partial derivatives of the function $f(x)=\|x\|^{\alpha}$ outside of $(0,0)$

We had the following solved problem in our lecture which I didn't quite fully understand. Find the partial derivatives of the function $f:\mathbb R^n\rightarrow \mathbb R$ $$f(x)=||x||^{\alpha}$$ ...
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4answers
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Understanding partial differential notation

Say i have $$z = x^2 + y$$ $$y = 2x$$ What sense does $\frac{\partial z}{\partial x}$ make? Can we talk about changing $z$ as $x$ changes but keeping $y$ constant?
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How can I solve this problem ? regarding partial derivative ? satisfying laplace euation

Show that if $$w = f(u,v) $$ satisfies the so called Laplace equation $$w_{uu} + w_{vv} = 0 $$and if $$u = \frac{(x^2 −y^2)}{2} , v = xy ,$$ then $$ w = w(x,y)$$ also satisfies the Laplace ...
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2answers
36 views

How can I express this solution in terms of the error function?

If I have this expression: $$u(x,t) = \frac {U_o}{\pi} \int_{-\infty}^{\infty} \!\frac{\sin(\alpha) \cos(\alpha x) e^{-k\alpha^2 t}}{\alpha} \,d\alpha, $$ how can I rewrite it in terms of the error ...
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2answers
35 views

Understanding partial derivative involving 3 variables

I am new to partial derivative and I need some help in understanding if what I have done so far is correct. Let $S$ be the surface given by $x^2 + y^2 - 3z^2 = 5$ I want to calculate the partial ...
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0answers
12 views

Integrating partial derivatives in a field equation

From some field equations, I was able to get the relation below. $$N(x,y) = \int \left[ \left( \frac{2}{x-2M(x,y)}\right)\frac{\partial M(x,y)}{\partial x}\right] dx $$. Now, I am trying to ...