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Интересная функция
![y= sqrt{1+x^2}*arctg(x)-ln(x+sqrt{1+x^2}) y= sqrt{1+x^2}*arctg(x)-ln(x+sqrt{1+x^2})](https://tex.z-dn.net/?f=y%3D+sqrt%7B1%2Bx%5E2%7D%2Aarctg%28x%29-ln%28x%2Bsqrt%7B1%2Bx%5E2%7D%29)
Производная
![y'= frac{2x}{2sqrt{1+x^2}}*arctg(x)+sqrt{1+x^2}* frac{1}{1+x^2} - frac{1}{x+sqrt{1+x^2}}*(1+frac{2x}{2sqrt{1+x^2}}) = y'= frac{2x}{2sqrt{1+x^2}}*arctg(x)+sqrt{1+x^2}* frac{1}{1+x^2} - frac{1}{x+sqrt{1+x^2}}*(1+frac{2x}{2sqrt{1+x^2}}) =](https://tex.z-dn.net/?f=y%27%3D+frac%7B2x%7D%7B2sqrt%7B1%2Bx%5E2%7D%7D%2Aarctg%28x%29%2Bsqrt%7B1%2Bx%5E2%7D%2A+frac%7B1%7D%7B1%2Bx%5E2%7D+-+frac%7B1%7D%7Bx%2Bsqrt%7B1%2Bx%5E2%7D%7D%2A%281%2Bfrac%7B2x%7D%7B2sqrt%7B1%2Bx%5E2%7D%7D%29+%3D)
![= frac{x*arctg(x)}{sqrt{1+x^2}}+ frac{1}{sqrt{1+x^2}} - frac{1}{x+sqrt{1+x^2}}*(1+frac{x}{sqrt{1+x^2}}) = = frac{x*arctg(x)}{sqrt{1+x^2}}+ frac{1}{sqrt{1+x^2}} - frac{1}{x+sqrt{1+x^2}}*(1+frac{x}{sqrt{1+x^2}}) =](https://tex.z-dn.net/?f=%3D+frac%7Bx%2Aarctg%28x%29%7D%7Bsqrt%7B1%2Bx%5E2%7D%7D%2B+frac%7B1%7D%7Bsqrt%7B1%2Bx%5E2%7D%7D+-+frac%7B1%7D%7Bx%2Bsqrt%7B1%2Bx%5E2%7D%7D%2A%281%2Bfrac%7Bx%7D%7Bsqrt%7B1%2Bx%5E2%7D%7D%29+%3D)
![= frac{x*arctg(x)}{sqrt{1+x^2}} + frac{1}{sqrt{1+x^2}} - frac{1}{x+sqrt{1+x^2}}*frac{sqrt{1+x^2}+x}{sqrt{1+x^2}} = = frac{x*arctg(x)}{sqrt{1+x^2}} + frac{1}{sqrt{1+x^2}} - frac{1}{x+sqrt{1+x^2}}*frac{sqrt{1+x^2}+x}{sqrt{1+x^2}} =](https://tex.z-dn.net/?f=%3D+frac%7Bx%2Aarctg%28x%29%7D%7Bsqrt%7B1%2Bx%5E2%7D%7D+%2B+frac%7B1%7D%7Bsqrt%7B1%2Bx%5E2%7D%7D+-+frac%7B1%7D%7Bx%2Bsqrt%7B1%2Bx%5E2%7D%7D%2Afrac%7Bsqrt%7B1%2Bx%5E2%7D%2Bx%7D%7Bsqrt%7B1%2Bx%5E2%7D%7D+%3D)
Производная
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