# 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 derivatives of a complex function

I have a problem considering the partial derivatives of a complex function. I am going to try to sketch the problem as best I can: I have a module, which takes as an input K and M (stiffness and mass ...
1 vote
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### Contradiction for $(dy/dx)^{-1}=dx/dy$

So I have seen in the derivation of Euler chain Equation (i.e. $\displaystyle \frac{\partial x}{\partial y} \cdotp \frac{\partial y}{\partial z} \cdotp \frac{\partial z}{\partial x} \ =\ -1$ ) where ...
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### A question about notation $\frac{\partial^2}{\partial\theta\partial\theta^T}l(\theta)$

For $l:\mathbb{R}^n\rightarrow\mathbb{R}$ a differentiable function and $\theta$ a vector, I read this notation $\frac{\partial^2}{\partial\theta\partial\theta^T}l(\theta)$ in a paper and want to ...
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### Is it possible to use product rule to solve $e^{i\theta_{y}}\dot{\theta_{y}}^{2}$ into the form which consists just $\dot{\theta_{y}}$

The formula in the title is incomplete due to character limit, here's the full form $$\frac{e^{i\theta_{y}}-e^{-i\theta_{y}}}{2i}\dot{\theta_{y}}^{2}$$ Into the form which has just $\dot{\theta_{y}}$ ...
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### Derivative of $\left\lvert y \right\rvert$ = $x^2$ with respect to x. [closed]

Pretty straight forward question that I need help with (This is part of a bigger question where |y|= $x^2$ is a curve, for which the tangent at a given point is to be found). Thanking you in advance.
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### Prove that there exists a smooth function $f$ defined on a neighborhood of $(0, 0)$ in $\mathbb{R}^2$ such that $f(0, 0) = 0$ and ...

Prove that there exists a smooth function $f$ defined on a neighborhood of $(0, 0)$ in $\mathbb{R}^2$ such that $f(0, 0) = 0$ and \begin{align*} \frac{\partial f}{\partial x} &= y e^{-x-y} - f, \\ ...
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### Determine whether this field is a gradient vector field

Let $n \in \mathbb{Z}$ and $X \colon \mathbb{R}^2\backslash\{(0,0)\} \rightarrow\mathbb{R}^2$ be the vector field $$X( x, y) = \begin{pmatrix}-y(x^2 + y^2)^n, x(x^2 + y^2)^n \end{pmatrix}.$$ a) For ...
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### Find the general solution of homogenous PDE

Find the solution of the equation $\partial^2{z/\partial{x^2}} + \partial^2{z/\partial{y^2}} = e^{-x}cosy$. I am able to find the Complementary Function as $z_c = φ_1(y + ix) + φ_2(y - ix)$. Please ...
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### Partial derivative on chain rule [closed]

Could anyone please guide me whether the solution of this partial derivative is correct? Solution from reference material: I have tried to calculate my own solution but it is different. My ...
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### Study the existence of the partial derivative and differential.

$$f(x, y) = \begin{cases} \frac{\sin(x^{1010}y^{1012})}{x^{2020} - y^{1010}x^{1010} + y^{2020}} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$ Hey! I have this ...
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