<span>Решение во вложении, надеюсь понятно</span>
∫(1/х^2+х^4)dx= ∫x^-2dx+∫x^4dx=-1/x+x^5/5+C
Если мы берём неопределенный интеграл,то добавляем нек-ую константу.Если определенный-не добавляем.
Х>8
9-x>0
- x >- 9
x < 9 (знак неравенства меняется в том случае, когда одно число(в нашем случае это -9) делится на отрицательное число(в нашем случае это -х, т.е. -1).
Получается система:
x>8
x<9
1) y = (2x² + 5x - 4)⁹
y' = 9(2x² + 5x - 4)⁸ * (2x² + 5x - 4)' = 9(2x² + 5x - 4)⁸ * (4x + 5) =
= 9(4x + 5)(2x² + 5x - 4)⁸
![2)y=\sqrt{2x^{2}-3x+7 }\\\\y'=\frac{1}{2\sqrt{2x^{2}-3x+7 } }*(2x^{2}-3x+7)'=\frac{4x-3}{2\sqrt{2x^{2}-3x+7 } }](https://tex.z-dn.net/?f=2%29y%3D%5Csqrt%7B2x%5E%7B2%7D-3x%2B7+%7D%5C%5C%5C%5Cy%27%3D%5Cfrac%7B1%7D%7B2%5Csqrt%7B2x%5E%7B2%7D-3x%2B7+%7D+%7D%2A%282x%5E%7B2%7D-3x%2B7%29%27%3D%5Cfrac%7B4x-3%7D%7B2%5Csqrt%7B2x%5E%7B2%7D-3x%2B7+%7D+%7D)
3) y = Sin3x
y' = Cos3x * (3x )' = 3Cos3x
4) y = Cos⁷x
y' = 7Cos⁶x * (Cosx)' = - 7Cos⁶xSinx
![5)y=e^{x^{7}-4 }\\\\y'=e^{x^{7}-4 }*(x^{7}-4)'=7x^{6}e^{x^{7} -4}](https://tex.z-dn.net/?f=5%29y%3De%5E%7Bx%5E%7B7%7D-4+%7D%5C%5C%5C%5Cy%27%3De%5E%7Bx%5E%7B7%7D-4+%7D%2A%28x%5E%7B7%7D-4%29%27%3D7x%5E%7B6%7De%5E%7Bx%5E%7B7%7D+-4%7D)
![6)y=ln(x^{3}+6)\\\\y'=\frac{1}{x^{3}+6 }*(x^{3}+6)'=\frac{3x^{2} }{x^{3}+6 }](https://tex.z-dn.net/?f=6%29y%3Dln%28x%5E%7B3%7D%2B6%29%5C%5C%5C%5Cy%27%3D%5Cfrac%7B1%7D%7Bx%5E%7B3%7D%2B6+%7D%2A%28x%5E%7B3%7D%2B6%29%27%3D%5Cfrac%7B3x%5E%7B2%7D+%7D%7Bx%5E%7B3%7D%2B6+%7D)
![7)y=\frac{1}{(x^{2} +4)^{3} }=(x^{2}+4)^{-3}\\\\y'=-3(x^{2}+4)^{-4}*(x^{2}+4)'=-\frac{3}{(x^{2}+4)^{4}}*2x=-\frac{6x}{(x^{2}+4)^{4} }](https://tex.z-dn.net/?f=7%29y%3D%5Cfrac%7B1%7D%7B%28x%5E%7B2%7D+%2B4%29%5E%7B3%7D+%7D%3D%28x%5E%7B2%7D%2B4%29%5E%7B-3%7D%5C%5C%5C%5Cy%27%3D-3%28x%5E%7B2%7D%2B4%29%5E%7B-4%7D%2A%28x%5E%7B2%7D%2B4%29%27%3D-%5Cfrac%7B3%7D%7B%28x%5E%7B2%7D%2B4%29%5E%7B4%7D%7D%2A2x%3D-%5Cfrac%7B6x%7D%7B%28x%5E%7B2%7D%2B4%29%5E%7B4%7D+%C2%A0%7D)
![8)y=lnSin2x\\\\y'=\frac{1}{Sin2x}*(Sin2x)'=\frac{1}{Sin2x}*Cos2x*(2x)'=\frac{2Cos2x}{Sin2x}=2Ctg2x](https://tex.z-dn.net/?f=8%29y%3DlnSin2x%5C%5C%5C%5Cy%27%3D%5Cfrac%7B1%7D%7BSin2x%7D%2A%28Sin2x%29%27%3D%5Cfrac%7B1%7D%7BSin2x%7D%2ACos2x%2A%282x%29%27%3D%5Cfrac%7B2Cos2x%7D%7BSin2x%7D%3D2Ctg2x)
9)y = Sin²(3x + 5)
y'= 2Sin(3x+5)*(Sin(3x + 5))' = 2Sin(3x + 5)Cos(3x + 5)*(3x + 5)' =
= 6Sin(3x + 5)Cos(3x + 5) = 3Sin(6x + 10)
![10)y=x^{2}*lnx^{2} \\\\y'=(x^{2})'*lnx^{2}+x^{2}*lnx^{2} =2xlnx^{2} +x^{2}*\frac{1}{x^{2} }*(x^{2})'= 2xlnx^{2}+\frac{x^{2} *2x}{x^{2} } =2xlnx^{2}+2x=2x(lnx^{2}+1)](https://tex.z-dn.net/?f=10%29y%3Dx%5E%7B2%7D%2Alnx%5E%7B2%7D+%5C%5C%5C%5Cy%27%3D%28x%5E%7B2%7D%29%27%2Alnx%5E%7B2%7D%2Bx%5E%7B2%7D%2Alnx%5E%7B2%7D+%3D2xlnx%5E%7B2%7D+%2Bx%5E%7B2%7D%2A%5Cfrac%7B1%7D%7Bx%5E%7B2%7D+%7D%2A%28x%5E%7B2%7D%29%27%3D+2xlnx%5E%7B2%7D%2B%5Cfrac%7Bx%5E%7B2%7D+%2A2x%7D%7Bx%5E%7B2%7D+%7D+%3D2xlnx%5E%7B2%7D%2B2x%3D2x%28lnx%5E%7B2%7D%2B1%29)