# Tagged Questions

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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### Proving $\det \big(Df\big|_x\big)=0$ for a function into unit circle

Let $f:\mathbb{R}^2\to S$ where $S=\{x\in\mathbb{R}^2:\, ||x||=1\}$. Prove that $\det \big(Df\big|_x\big)=0$ for all $x$. I'm having trouble attacking this. So I need to show that there is some ...
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### Monotonic and differentiable function

Question: $f: R\to R$ is a differentiable and monotonic function such that $f(f(x)) = k(x^{11} + x), (k \neq 0)$. Find the values that $k$ can take. Differentiating the given expression: ...
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### ${\partial\over{\partial x_j}}\left(\partial x_i\over\partial t\right)\ne{\partial\over{\partial t}}\left(\partial x_i\over\partial x_j\right)$?

$\boldsymbol x = f(\boldsymbol X,t)$ is the position of a particle in an instant of time $\boldsymbol X$ is the initial position $t$ time $\boldsymbol u$ velocity In my opnion $f$ is continuos... ...
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### The composition Baire class one with a differentiable function

It is known that derivative of differentiable function is Baire class one and it is also known that composition of two Baire class one functions may not Baire one class one fuction. Let $f$ be a ...
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### Help Me Understand How this was Derived

Here is the question: "Newton’s Law of Cooling. Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and ...
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### A specific space curve's length

I'm trying to calculate a space curve's length. $$r(t)=(3t^2-2t, t^3, 1-t)$$ $t$: from $0$ to $2$ So I have to derivate the $r(t)$, which makes: $$r'(t)=(6t-2, 3t^2, -1)$$ And then I get the ...
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### How many non-differentiable functions exist?

The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?
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### Derivative with respect to a derivative

Let $q=q(t)\in C^1(\mathbb{R})$ and $V=V(x)\in C^1(\mathbb{R})$. My book uses the following fact over and over again $$\frac{\partial V(q)}{\partial \dot{q}}=0.$$ Why is this true?
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### How to show that a function is differentiable even though its partial derivatives in origin don't exist

I have a function $f(x,y) = \begin{cases} (x^2+y^2)\sin(\frac{1}{x^2+y^2}), & (x,y)\neq(0,0) \\ 0, & (x,y)=(0,0) \end{cases}$ and I need to show that $f(x,y)$ is differentiable, even though ...
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### Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, ...
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### Showing that function of a function is differentiable

Let f be entire and define g(z)=arg(z)f(z). Prove that g is differentiable at w if and only if f(w)=0. Not really sure how to go about this?
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### Showing $\frac{d^2}{dr^2}\left(\frac{1}{r}\frac{d}{dr}\right)^{k-1}(r^{2k-1}\phi(r))=\left(\frac{1}{r}\frac{d}{dr}\right)^{k}(r^{2k}\phi'(r))$

How to show that $\frac{d^2}{dr^2}\left(\frac{1}{r}\frac{d}{dr}\right)^{k-1}(r^{2k-1}\phi(r))=\left(\frac{1}{r}\frac{d}{dr}\right)^{k}(r^{2k}\phi'(r))$ for $k\ge 1, r>0$ and $\phi$ sufficiently ...
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### If $x^2+ax-3x-(a+2)=0\;,$ Then $\min\left(\frac{a^2+1}{a^2+2}\right)$

If $x^2+ax-3x-(a+2)=0\;,$ Then $\displaystyle \min\left(\frac{a^2+1}{a^2+2}\right)$ $\bf{My\; Try::}$ Given $x^2+ax-3x-(a+2)=0\Leftrightarrow ax-a = -(x^2-3x-2)$ So we get ...
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### Shortest distance between two objects moving along two lines

I've got two objects defined by a position vector and a velocity vector. I want to know how close they will come so I can implement avoidance behaviour. This all as to be done by and algorithm. ...
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### Separation of variables for $tu_t = u_{xx} + 2u$

Separate the variables for the equation $$tu_t = u_{xx} + 2u$$ with the boundary conditions $u(0,t) = u(π,t) = 0$. Show that there are an inﬁnite number of solutions which satisfy the initial ...
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### Derivative of piecewise functions

I was going through some solved examples when I came across this sum. My doubt is that while calculating the derivative of the function at 0 why has the right hand derivative (that is the right hand ...
### how to solve this differential equation $(\sin y+2x\cos^2 y)\ dx+x\cos y(2x \sin y+1)\ dy=0$
This is my first question here. I tried to solve this ODE. $$\left(\sin(y)+2x \cos^2(y)\right) \mathrm{d}x + x\cos(y) \left(2x \sin(y)+1\right) \mathrm{d}y = 0$$ Any suggestion?
### What's the second Fréchet derivative of a function $\mathbb R^d\to\mathbb R$
Let $u:\mathbb R^d\to\mathbb R$ be twice Fréchet differentiable. What's the second Fréchet derivative ${\rm D}^2u$ of $u$? It's clear that ${\rm D}u$ is a mapping$^1$ \$\mathbb R^d\to\mathfrak ...