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Bookmark or Share- Limit of $\\frac{x^c-c^x}{x^x-c^c}$ as $x \\rightarrow c$My question is: Show that $\lim_{x \rightarrow c} \frac{x^c-c^x}{x^x-c^c}$ exists and find its value. Because the limit is 0/0 I've tried using L'Hopital's rule, but every time I differentiate it I
- Integral: $\\int_{-\\infty}^{\\infty} x^2 e^{-x^2}\\mathrm dx$I don't know how to evaluate it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates. $$\\int_{-\\infty}^\\infty x^2 e^...
- Derivative of sigmoid function $\\sigma (x) = \\frac{1}{1+e^{-x}}$In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\\sigma (x) = \\frac{1}{1+e^{-x}}.$$ This function is easy to differentiate
- solving the integral of $e^{x^2}$ - Mathematics Stack ExchangeStack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
- How to prove $dxdy = r dr d \\theta$? - Mathematics Stack ExchangeA piece of an annulus swept out by a change of angle $\Delta \theta$ and a change of radius $\Delta r$, starting from a point given by $(r,\theta)$, has area $\Delta \theta \int_r^{r+\Delta r} s ds = \Delta \theta \frac{(r+\Delta r)^2-r^2}{2} = \Delta \theta \left ( r \Delta r + \frac{\Delta r^2}{2} \right )$.
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