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y = ( 2 + 5 x ) 10 x {\displaystyle y={\frac {(2+5x)^{10}}{x}}}
y ′ = [ ( 2 + 5 x ) 10 x ] ′ = − 1 x 2 ( 2 + 5 x ) 10 + 1 x ⋅ 10 ( 2 + 5 x ) 9 ⋅ 5 = − ( 2 + 5 x ) 10 x 2 + 50 ( 2 + 5 x ) 9 x {\displaystyle {\begin{aligned}y'&=\left[{\frac {(2+5x)^{10}}{x}}\right]'\\&=-{\frac {1}{x^{2}}}(2+5x)^{10}+{\frac {1}{x}}\cdot 10(2+5x)^{9}\cdot 5\\&=-{\frac {(2+5x)^{10}}{x^{2}}}+{\frac {50(2+5x)^{9}}{x}}\end{aligned}}}
y = x cos ( 2 x + π 2 ) sin ( 2 x + π 2 ) {\displaystyle y=x\cos \left(2x+{\frac {\pi }{2}}\right)\sin \left(2x+{\frac {\pi }{2}}\right)}
y = x ( − sin 2 x ) cos 2 x = − 1 2 x sin 4 x {\displaystyle y=x(-\sin 2x)\cos 2x=-{\frac {1}{2}}x\sin 4x}
y ′ = ( − 1 2 x sin 4 x ) ′ = − 1 2 sin 4 x − 1 2 x cos 4 x ⋅ 4 = − 1 2 sin 4 x − 2 x cos 4 x {\displaystyle {\begin{aligned}y'&=\left(-{\frac {1}{2}}x\sin 4x\right)'\\&=-{\frac {1}{2}}\sin 4x-{\frac {1}{2}}x\cos 4x\cdot 4\\&=-{\frac {1}{2}}\sin 4x-2x\cos 4x\\\end{aligned}}}
![{\displaystyle {\begin{aligned}y'&=\left[{\frac {(2+5x)^{10}}{x}}\right]'\\&=-{\frac {1}{x^{2}}}(2+5x)^{10}+{\frac {1}{x}}\cdot 10(2+5x)^{9}\cdot 5\\&=-{\frac {(2+5x)^{10}}{x^{2}}}+{\frac {50(2+5x)^{9}}{x}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e394ae322b224115296818bf5d083efbd30d809b)
