# Tagged Questions

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### Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
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### Is there an actual function that “escapes” from zero after an amount of derivatives?

Suppose we have a function $f$ such that for all $i=2,...k: f^{(i)}=0$ but for $i\ge k+1$ we have $f^{(i)}\neq 0$. Can there be such functions in theory ? and is there an actual function that ...
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### $f'(x)$ to $f(x)$ is it possible without knowing the value of $f'(x)$ or $f(x)$

I don't know much math and i got stuck at a problem: I'm not sure if it possible how to do it. I must use hermite interpolation for the following: 'Find the polynomial interpolating the function $f$ ...
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### Reciprocal Rule Question

$f(z) = \frac{4 z + 4}{z^2 + z + 1}$ I have a question about the reciprocal rule concerning derivatives. I want to work around doing the quotient rule for the above function if possible, but I am not ...
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### Find fourth derivative of function

The function is $\displaystyle{\frac{3x^4}{1-x}}$ and I am trying to find $\displaystyle{\frac{d^4}{dx^4}}$. However, I want to find the solution without using the quotient rule $4$ times in a row. I ...
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### I can't seem to find this derivative any help would be great.

A rocket of mass m = 1000 kg is traveling in a straight line for a short time. The distance in meters covered by the rocket during this time is described by the function $r(t)=t^3 âˆ’3t^2 +6t$ where ...
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### If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
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### Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
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### A question on a multivariable continuously differentiable function

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that $x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect ...