Предмет: Алгебра,
автор: Sharmik
Найдите производные следующие функции:
1) y=(3-2x^3 )^5
2) y=(x^4-x-1)^4
3) y= √(x^3+1)
4) y= ^3√((1-x^2 )^2 )
5) y=2*cos5x
6) y=ln(sinx)
7) y=ln^3(x^2-1)
8) y = ln(ln^35x)
Ответы
Автор ответа:
0
1)![(3-2x^{3})^{5}=5(3-2x^{3})^{4}((3-2x^{3}))'=-10(3-2x^{3})^{4}*( x^{3} )' (3-2x^{3})^{5}=5(3-2x^{3})^{4}((3-2x^{3}))'=-10(3-2x^{3})^{4}*( x^{3} )'](https://tex.z-dn.net/?f=%283-2x%5E%7B3%7D%29%5E%7B5%7D%3D5%283-2x%5E%7B3%7D%29%5E%7B4%7D%28%283-2x%5E%7B3%7D%29%29%27%3D-10%283-2x%5E%7B3%7D%29%5E%7B4%7D%2A%28+x%5E%7B3%7D+%29%27)
![=-30x^{2}(3-2x^{3})^{4} =-30x^{2}(3-2x^{3})^{4}](https://tex.z-dn.net/?f=%3D-30x%5E%7B2%7D%283-2x%5E%7B3%7D%29%5E%7B4%7D)
2)![=4(x^{4}-x-1)^{3}*(x^{4}-x-1)'=4(x^{4}-x-1)^{3}*(4 x^{3} -1) =4(x^{4}-x-1)^{3}*(x^{4}-x-1)'=4(x^{4}-x-1)^{3}*(4 x^{3} -1)](https://tex.z-dn.net/?f=%3D4%28x%5E%7B4%7D-x-1%29%5E%7B3%7D%2A%28x%5E%7B4%7D-x-1%29%27%3D4%28x%5E%7B4%7D-x-1%29%5E%7B3%7D%2A%284+x%5E%7B3%7D+-1%29)
3)![(sqrt{x^{3}+1})'= frac{3x^{2}}{2 sqrt{x^{3}+1} } (sqrt{x^{3}+1})'= frac{3x^{2}}{2 sqrt{x^{3}+1} }](https://tex.z-dn.net/?f=+%28sqrt%7Bx%5E%7B3%7D%2B1%7D%29%27%3D+frac%7B3x%5E%7B2%7D%7D%7B2++sqrt%7Bx%5E%7B3%7D%2B1%7D+%7D++)
4)![= frac{(1- x^{2} )^{2}'}{3((1- x^{2} )^{2})^ frac{2}{3} } = frac{2(1- x^{2} )*(1- x^{2} )'}{3((1- x^{2} )^{2})^ frac{2}{3} } = = frac{(1- x^{2} )^{2}'}{3((1- x^{2} )^{2})^ frac{2}{3} } = frac{2(1- x^{2} )*(1- x^{2} )'}{3((1- x^{2} )^{2})^ frac{2}{3} } =](https://tex.z-dn.net/?f=%3D+frac%7B%281-+x%5E%7B2%7D+%29%5E%7B2%7D%27%7D%7B3%28%281-+x%5E%7B2%7D+%29%5E%7B2%7D%29%5E+frac%7B2%7D%7B3%7D+%7D+%3D+frac%7B2%281-+x%5E%7B2%7D+%29%2A%281-+x%5E%7B2%7D+%29%27%7D%7B3%28%281-+x%5E%7B2%7D+%29%5E%7B2%7D%29%5E+frac%7B2%7D%7B3%7D+%7D+%3D)
![= frac{4x(1- x^{2} )}{3((1- x^{2} )^{2})^ frac{2}{3} } = frac{4x(1- x^{2} )}{3((1- x^{2} )^{2})^ frac{2}{3} }](https://tex.z-dn.net/?f=%3D+frac%7B4x%281-+x%5E%7B2%7D+%29%7D%7B3%28%281-+x%5E%7B2%7D+%29%5E%7B2%7D%29%5E+frac%7B2%7D%7B3%7D+%7D+)
5)![(2cos5x)'=-10sin5x (2cos5x)'=-10sin5x](https://tex.z-dn.net/?f=%282cos5x%29%27%3D-10sin5x)
6)![ln(sinx)'=? ln(sinx)'=?](https://tex.z-dn.net/?f=ln%28sinx%29%27%3D%3F)
7)![ln^{3}( x^{2} -1)'=3ln^{2}( x^{2} -1)*ln( x^{2} -1)'= ln^{3}( x^{2} -1)'=3ln^{2}( x^{2} -1)*ln( x^{2} -1)'=](https://tex.z-dn.net/?f=ln%5E%7B3%7D%28+x%5E%7B2%7D+-1%29%27%3D3ln%5E%7B2%7D%28+x%5E%7B2%7D+-1%29%2Aln%28+x%5E%7B2%7D+-1%29%27%3D)
![=3ln^{2}( x^{2} -1)* frac{( x^{2} -1)'}{ x^{2} -1} =frac{6x*ln^{2}( x^{2} -1) }{ x^{2} -1} =3ln^{2}( x^{2} -1)* frac{( x^{2} -1)'}{ x^{2} -1} =frac{6x*ln^{2}( x^{2} -1) }{ x^{2} -1}](https://tex.z-dn.net/?f=%3D3ln%5E%7B2%7D%28+x%5E%7B2%7D+-1%29%2A+frac%7B%28+x%5E%7B2%7D+-1%29%27%7D%7B+x%5E%7B2%7D+-1%7D+%3Dfrac%7B6x%2Aln%5E%7B2%7D%28+x%5E%7B2%7D+-1%29+%7D%7B+x%5E%7B2%7D+-1%7D+)
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