# Tagged Questions

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### How do I derive this recursive function?

I have a function $S_t = a*Y_t+(1-a)*S_{t-1}$ Where $a\in (0,1)\cap \mathbb{Q}$, $t$ represents a unit of time $Y_t$ is the value at a time period $t$ I am trying to find $dS_t \over dt$ I have ...
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### Derivative of a function when it is squared.

Was wondering when you are for example finding $dw/dt$ but you are given a function like $w^2(t)=r^2-2\cos(t)$, when r is some constant, how you are supposed to solve it? Are you supposed to ...
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### Partial derivatives exist, but the function is not differentiable

It is well-known that a function $f:\mathbb{R}^n\to \mathbb{R}$ can have the property that it is differentiable along any line through the origin, but not even continuous at the origin. Can the same ...
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### Properties and notation of third-order (and higher) partial-derivatives

This question has been bothering me for quite a while and I still haven't found a satisfying answer anywhere on the internet or in any of my books (which may not be that advanced, mind you...). Since ...
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### What is the derivative of $\dot{x} = f(x(t))$?

I am supposed to take the derivative of a function similar to this one: Take the derivative of $$\dot{x} = \cos(x)$$ where $x$ is a function of $t.$ I believe that this can be generalized to the ...
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### Is my calculation right for differentiability?(with complete resolution if right)

In the following completed example I ask if it is done right. $$f(x,y)=\begin{cases} \frac {2x^2y}{x^2+y^2} \mbox{for} (x,y)\neq (0,0) \\0 \mbox{for} (x,y)=(0,0) \end{cases}$$ Now and the partial ...
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### Question on Partial Derviatives

For function $f(x,y) = x^2 y$ The partial derivatives for $x$ is $2.x.y$. I'm new to such math equation and i'm learning them now. May i know why is it so? Thanks!
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### Second derivative expression

I have $f:\mathbb R^n\to \mathbb R$ and $\gamma:\mathbb R \to \mathbb R^n$, which are both $\mathrm C^2$. Considering $g=f\circ \gamma$, how could I express $g''$, second derivative of $g$ in terms of ...
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### Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
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### Prove that $f$ is differentiable in $(0,0)$ if and only if $\lim_{t\to0+} g(t)$ exists

Let $g:[0,\infty)\to\mathbb{R}$ be a mapping and $f(x,y)=xg(\sqrt{x^2+y^2})$ for all $(x,y)\in\mathbb{R^2}$. Prove that $f$ is differentiable in $(0,0)$ $\iff$ $\lim_{t\to0+} g(t)$ exists. My ...
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### Verification: Hessian of the following composition.

I was hoping that someone could verify the steps of computing a Hessian matrix. I have the following function, $F:\mathbb{R}^n\to\mathbb{R}$, $$F({\bf x}) = \sum_{i=1}^mf(g(A_i^T{\bf x}))$$ where ...
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### Direction for greatest derivative

Suppose I have a function like $f(x,y) = e^x e^y x^2 y^2$, and I want to know in which direction the derivative will grow fastest at a stationary point. $(0,0)$ is a stationary point of the example ...
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### What is the 1st derivative i.r.t. coordinates for a vector function?

For a vector function $f(x,y,z)$, we have the divergence $$\nabla \cdot f(x,y,z) = \frac{\partial{f}_{x}}{\partial x}+\frac{\partial{f}_{y}}{\partial y}+\frac{\partial{f}_{z}}{\partial z}$$ , the ...
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### Find the partial derivatives of second order of $f(x,y)=\varphi(xy,\frac{x}{y})$

Ok guys, I'm given this smooth function $\varphi(u,v)$ defined in $R^2$. So that $f(x,y)=\varphi(xy,\frac{x}{y})$. I have to find all partial derivatives of second order of $f$ using the partial ...