ПОМОГИТЕ, ПОЖАЛУЙСТА!!! ОЧЕНЬ НАДО. (ну или, кто что сможет)
определить производные функции:
а) у=ln(3x²+![sqrt{9 x^{4} +1} ) sqrt{9 x^{4} +1} )](https://tex.z-dn.net/?f=+sqrt%7B9+x%5E%7B4%7D+%2B1%7D+%29)
б) y=x*10 в степени ![sqrt{x} sqrt{x}](https://tex.z-dn.net/?f=+sqrt%7Bx%7D+)
в) xy+
Ответы
Ответ дал:
0
a)
![y=ln(3x ^{2} + \sqrt{9 x^{4} +1} ) y=ln(3x ^{2} + \sqrt{9 x^{4} +1} )](https://tex.z-dn.net/?f=y%3Dln%283x+%5E%7B2%7D+%2B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%29)
По формуле
![(lnu)`= \frac{1}{u} \cdot u` (lnu)`= \frac{1}{u} \cdot u`](https://tex.z-dn.net/?f=%28lnu%29%60%3D+%5Cfrac%7B1%7D%7Bu%7D+%5Ccdot+u%60)
![y`= \frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (3x ^{2} + \sqrt{9 x^{4} +1} )`= \\ \\ = \frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (2x + \frac{1}{2 \sqrt{9 x^{4} +1} }\cdot(9 x^{4} +1})`)= \\ \\ = \frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (2x + \frac{1}{2 \sqrt{9 x^{4} +1} }\cdot36 x^{3} })= \\ \\ = y`= \frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (3x ^{2} + \sqrt{9 x^{4} +1} )`= \\ \\ = \frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (2x + \frac{1}{2 \sqrt{9 x^{4} +1} }\cdot(9 x^{4} +1})`)= \\ \\ = \frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (2x + \frac{1}{2 \sqrt{9 x^{4} +1} }\cdot36 x^{3} })= \\ \\ =](https://tex.z-dn.net/?f=y%60%3D+%5Cfrac%7B1%7D%7B3x+%5E%7B2%7D+%2B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D%5Ccdot+%283x+%5E%7B2%7D+%2B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%29%60%3D++%5C%5C++%5C%5C+%3D+%5Cfrac%7B1%7D%7B3x+%5E%7B2%7D+%2B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D%5Ccdot+%282x++%2B++%5Cfrac%7B1%7D%7B2+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D%5Ccdot%289+x%5E%7B4%7D+%2B1%7D%29%60%29%3D+%5C%5C++%5C%5C+%3D++%5Cfrac%7B1%7D%7B3x+%5E%7B2%7D+%2B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D%5Ccdot+%282x++%2B++%5Cfrac%7B1%7D%7B2+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D%5Ccdot36+x%5E%7B3%7D+%7D%29%3D++%5C%5C++%5C%5C+%3D)
![=\frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (2x + \frac{18x^{3} }{ \sqrt{9 x^{4} +1} } }) =\frac{1}{3x ^{2} + \sqrt{9 x^{4} +1} }\cdot (2x + \frac{18x^{3} }{ \sqrt{9 x^{4} +1} } })](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B3x+%5E%7B2%7D+%2B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D%5Ccdot+%282x++%2B++%5Cfrac%7B18x%5E%7B3%7D+%7D%7B+%5Csqrt%7B9+x%5E%7B4%7D+%2B1%7D+%7D+%7D%29)
б)
![y`=(x)`\cdot10 ^{ \sqrt{x} } +x\cdot (10 ^{ \sqrt{x} })`= \\ \\ = 1\cdot10 ^{ \sqrt{x} } +x\cdot (10 ^{ \sqrt{x} })\cdot ln10\cdot ( \sqrt{x} )`= \\ \\ = 10 ^{ \sqrt{x} } +x\cdot (10 ^{ \sqrt{x} })\cdot ln10\cdot ( \frac{1}{2 \sqrt{x}} )= \\ \\= 10 ^{ \sqrt{x} } + \frac{ \sqrt{x} }{2} \cdot (10 ^{ \sqrt{x} })\cdot ln10= \\ \\ y`=(x)`\cdot10 ^{ \sqrt{x} } +x\cdot (10 ^{ \sqrt{x} })`= \\ \\ = 1\cdot10 ^{ \sqrt{x} } +x\cdot (10 ^{ \sqrt{x} })\cdot ln10\cdot ( \sqrt{x} )`= \\ \\ = 10 ^{ \sqrt{x} } +x\cdot (10 ^{ \sqrt{x} })\cdot ln10\cdot ( \frac{1}{2 \sqrt{x}} )= \\ \\= 10 ^{ \sqrt{x} } + \frac{ \sqrt{x} }{2} \cdot (10 ^{ \sqrt{x} })\cdot ln10= \\ \\](https://tex.z-dn.net/?f=y%60%3D%28x%29%60%5Ccdot10+%5E%7B+%5Csqrt%7Bx%7D+%7D+%2Bx%5Ccdot+%2810+%5E%7B+%5Csqrt%7Bx%7D+%7D%29%60%3D+%5C%5C++%5C%5C+%3D+1%5Ccdot10+%5E%7B+%5Csqrt%7Bx%7D+%7D+%2Bx%5Ccdot+%2810+%5E%7B+%5Csqrt%7Bx%7D+%7D%29%5Ccdot+ln10%5Ccdot+%28+%5Csqrt%7Bx%7D+%29%60%3D+%5C%5C++%5C%5C+%3D++10+%5E%7B+%5Csqrt%7Bx%7D+%7D+%2Bx%5Ccdot+%2810+%5E%7B+%5Csqrt%7Bx%7D+%7D%29%5Ccdot+ln10%5Ccdot+%28+%5Cfrac%7B1%7D%7B2++%5Csqrt%7Bx%7D%7D+%29%3D+%5C%5C++%5C%5C%3D++10+%5E%7B+%5Csqrt%7Bx%7D+%7D+%2B+%5Cfrac%7B+%5Csqrt%7Bx%7D+%7D%7B2%7D+%5Ccdot+%2810+%5E%7B+%5Csqrt%7Bx%7D+%7D%29%5Ccdot+ln10%3D+%5C%5C++%5C%5C++)
в)
![(xy+ e^{y})`=0 \\ \\ x`\cdot y+x\cdot y`+e ^{y}\cdot y`=0 \\ \\ y+x\cdot y`+e ^{y}\cdot y`=0 \\ \\y`= -\frac{y}{x+e ^{y} } (xy+ e^{y})`=0 \\ \\ x`\cdot y+x\cdot y`+e ^{y}\cdot y`=0 \\ \\ y+x\cdot y`+e ^{y}\cdot y`=0 \\ \\y`= -\frac{y}{x+e ^{y} }](https://tex.z-dn.net/?f=%28xy%2B+e%5E%7By%7D%29%60%3D0+%5C%5C++%5C%5C+x%60%5Ccdot+y%2Bx%5Ccdot+y%60%2Be+%5E%7By%7D%5Ccdot+y%60%3D0++%5C%5C++%5C%5C+y%2Bx%5Ccdot+y%60%2Be+%5E%7By%7D%5Ccdot+y%60%3D0+%5C%5C+++%5C%5Cy%60%3D+-%5Cfrac%7By%7D%7Bx%2Be+%5E%7By%7D+%7D+)
По формуле
б)
в)
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