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производная сложно-показательной функции:
1)логарифмируем левую и правую часть функции:
2) упрощаем
3) берем производную от левой и правой частей функции
4) выражаем y'
![y=(lnx+e^x)^{ \sqrt{x} } \\ \\ lny=ln[(lnx+e^x)^{ \sqrt{x} }] \\ \\lny=\sqrt{x} \ ln[(lnx+e^x) \\ \\ (lny)'=(\sqrt{x} )' \ ln[(lnx+e^x)] +(ln[(lnx+e^x)])' \sqrt{x} \\ \\ \frac{1}{y} *y'= \frac{1}{2\sqrt{x}} \ ln[(lnx+e^x)]+ \frac{1}{lnx+e^x} *( \frac{1}{x}+e^x) \ \sqrt{x} \\ \\ y'=y \ [\frac{1}{2\sqrt{x}} \ ln[(lnx+e^x)]+ \frac{1}{lnx+e^x} *( \frac{1}{x}+e^x) \ \sqrt{x} \ ] \\](https://tex.z-dn.net/?f=y%3D%28lnx%2Be%5Ex%29%5E%7B+%5Csqrt%7Bx%7D+%7D+%5C%5C+%5C%5C+lny%3Dln%5B%28lnx%2Be%5Ex%29%5E%7B+%5Csqrt%7Bx%7D+%7D%5D+%5C%5C+%5C%5Clny%3D%5Csqrt%7Bx%7D+%5C+ln%5B%28lnx%2Be%5Ex%29+%5C%5C+%5C%5C+%28lny%29%27%3D%28%5Csqrt%7Bx%7D+%29%27+%5C+ln%5B%28lnx%2Be%5Ex%29%5D+%2B%28ln%5B%28lnx%2Be%5Ex%29%5D%29%27+%5Csqrt%7Bx%7D++%5C%5C++%5C%5C++%5Cfrac%7B1%7D%7By%7D+%2Ay%27%3D+%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D+%5C++ln%5B%28lnx%2Be%5Ex%29%5D%2B+%5Cfrac%7B1%7D%7Blnx%2Be%5Ex%7D+%2A%28+%5Cfrac%7B1%7D%7Bx%7D%2Be%5Ex%29+%5C+%5Csqrt%7Bx%7D+%5C%5C+%5C%5C+y%27%3Dy+%5C+%5B%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D+%5C++ln%5B%28lnx%2Be%5Ex%29%5D%2B+%5Cfrac%7B1%7D%7Blnx%2Be%5Ex%7D+%2A%28+%5Cfrac%7B1%7D%7Bx%7D%2Be%5Ex%29+%5C+%5Csqrt%7Bx%7D+%5C+%5D+%5C%5C+)
Так как
то конечный ответ:
![y'=(lnx+e^x)^{ \sqrt{x} } \ [\frac{1}{2\sqrt{x}} \ ln[(lnx+e^x)]+ \frac{1}{lnx+e^x} *( \frac{1}{x}+ e^x) \ \sqrt{x} \ ] = \\ \\ = y'=(lnx+e^x)^{ \sqrt{x} } \ [\frac{ln[(lnx+e^x)]}{2\sqrt{x}} \ + \frac{1}{lnx+e^x} *( \frac{ \sqrt{x}}{x}+ e^x) \ ]](https://tex.z-dn.net/?f=y%27%3D%28lnx%2Be%5Ex%29%5E%7B+%5Csqrt%7Bx%7D+%7D+%5C+%5B%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D+%5C++ln%5B%28lnx%2Be%5Ex%29%5D%2B+%5Cfrac%7B1%7D%7Blnx%2Be%5Ex%7D+%2A%28+%5Cfrac%7B1%7D%7Bx%7D%2B+e%5Ex%29+%5C+%5Csqrt%7Bx%7D+%5C+%5D+%3D+%5C%5C+%5C%5C+%3D+y%27%3D%28lnx%2Be%5Ex%29%5E%7B+%5Csqrt%7Bx%7D+%7D+%5C+%5B%5Cfrac%7Bln%5B%28lnx%2Be%5Ex%29%5D%7D%7B2%5Csqrt%7Bx%7D%7D+%5C++%2B+%5Cfrac%7B1%7D%7Blnx%2Be%5Ex%7D+%2A%28+%5Cfrac%7B+%5Csqrt%7Bx%7D%7D%7Bx%7D%2B+e%5Ex%29+%5C+%5D)
![OTBET: \ y'=(lnx+e^x)^{ \sqrt{x} } \ [\frac{ln[(lnx+e^x)]}{2\sqrt{x}} \ + \frac{1}{lnx+e^x} *( \frac{ \sqrt{x}}{x}+ e^x) \ ]](https://tex.z-dn.net/?f=OTBET%3A+%5C+y%27%3D%28lnx%2Be%5Ex%29%5E%7B+%5Csqrt%7Bx%7D+%7D+%5C+%5B%5Cfrac%7Bln%5B%28lnx%2Be%5Ex%29%5D%7D%7B2%5Csqrt%7Bx%7D%7D+%5C++%2B+%5Cfrac%7B1%7D%7Blnx%2Be%5Ex%7D+%2A%28+%5Cfrac%7B+%5Csqrt%7Bx%7D%7D%7Bx%7D%2B+e%5Ex%29+%5C+%5D)
1)логарифмируем левую и правую часть функции:
2) упрощаем
3) берем производную от левой и правой частей функции
4) выражаем y'
Так как
то конечный ответ: