# Tagged Questions

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### Determine $\frac{\partial}{\partial r} \int\!\!\!\!\!\!-_{B(x,r)} \frac{r}{n} u_{xx}(y)dy$

I'd like to show $\frac{\partial}{\partial r} \int\!\!\!\!\!\!-_{B(x,r)} \frac{r}{n} u_{xx}(y)dy = \int\!\!\!\!\!\!-_{\partial B(x,r)}u_{xx}dS + (\frac{1}{n}-1)\int\!\!\!\!\!\!-_{B(x,r)} u_{xx} dy$ ...
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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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### Non zero partial derivatives in implicit function theorem?

Simple proofs of this in 3 dimensions $z(x,y)$ impose a $dz = 0$ constraint to solve for $dy/dx$. This implies $z(x, y) = \text{constant}$. Why are the partial derivatives in general in the theorem ...
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### condition for differentiability

Lets say you have a function: $$f: \mathbb{R}^2 \rightarrow \mathbb{R^2}=((u(x,y),v(x,y)).$$ Does it follow directly from this definition: ...
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### Is this an ordinary differential equation?

If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable it is said to be an ordinary differential equation (ODE). If ...
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### Derivative of Unknown Function

Let's say I have a function $c(t), t \in \mathbb{R}$ and I don't know anything about it other than it is a function of $t$. If I derive said function with ... $x$ for example, what is the result? ...
Suppose that $D\subset\mathbb{R}^m$ and $g(\cdot)$ is a smooth function mapping $D$ into $\mathbb{R}$ with a unique minimum at $\hat{x}$ lying in the interior of $D$. Then, the Laplace's approximation ...