Номер 20, страница 46, часть 2 - гдз по алгебре 10 класс учебник Пак, Ардакулы

20. Найдите производные следующих функций:

а)

$f(x)=2\sin\frac{x}{2}$, $g(x)=\frac{7}{3}\sin 3x$, $h(x)=-\frac{1}{7}\sin\left(14x+\frac{\pi}{6}\right)$,

$u(x)=\frac{1}{5\pi}\sin\left(\frac{\pi}{8}-10\pi x\right)$;

б)

$f(x)=-22\cos\frac{x}{11}$, $g(x)=\frac{1}{18}\cos 27x$, $h(x)=-\frac{2}{9}\cos\left(\frac{3x}{2}-\frac{\pi}{7}\right)$,

$u(x)=-\frac{8}{9}\cos\left(\frac{\pi}{8}-\frac{27\pi x}{16}\right)$;

в)

$f(x)=4\tan\frac{x}{4}$, $g(x)=\frac{3}{4}\tan\frac{8x}{9}$, $h(x)=\frac{4}{45}\tan\left(\frac{15x}{2}-\frac{\pi}{9}\right)$,

$u(x)=\frac{1}{2\pi}\tan\left(\frac{\pi}{5}-5\pi x\right)$;

г)

$f(x)=-\frac{1}{3}\cot\frac{x}{3}$, $g(x)=\frac{9}{11}\cot\frac{22x}{3}$, $h(x)=\frac{3}{5}\cot\left(\frac{10x}{9}-\frac{\pi}{21}\right)$,

$u(x)=-\frac{1}{25\pi}\cot\left(\frac{\pi}{11}-75\pi x\right)$.

а)

$f(x)=2\sin\frac{x}{2}$
$f'(x) = (2\sin\frac{x}{2})' = 2 \cdot \cos(\frac{x}{2}) \cdot (\frac{x}{2})' = 2 \cos(\frac{x}{2}) \cdot \frac{1}{2} = \cos(\frac{x}{2})$.
Ответ: $f'(x) = \cos(\frac{x}{2})$.

$g(x) = \frac{7}{3}\sin 3x$
$g'(x) = (\frac{7}{3}\sin 3x)' = \frac{7}{3} \cdot \cos(3x) \cdot (3x)' = \frac{7}{3} \cos(3x) \cdot 3 = 7\cos(3x)$.
Ответ: $g'(x) = 7\cos(3x)$.

$h(x)=-\frac{1}{7}\sin(14x+\frac{\pi}{6})$
$h'(x) = (-\frac{1}{7}\sin(14x+\frac{\pi}{6}))' = -\frac{1}{7} \cdot \cos(14x+\frac{\pi}{6}) \cdot (14x+\frac{\pi}{6})' = -\frac{1}{7} \cos(14x+\frac{\pi}{6}) \cdot 14 = -2\cos(14x+\frac{\pi}{6})$.
Ответ: $h'(x) = -2\cos(14x+\frac{\pi}{6})$.

$u(x) = \frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x)$
$u'(x) = (\frac{1}{5\pi}\sin(\frac{\pi}{8}-10\pi x))' = \frac{1}{5\pi} \cdot \cos(\frac{\pi}{8}-10\pi x) \cdot (\frac{\pi}{8}-10\pi x)' = \frac{1}{5\pi} \cos(\frac{\pi}{8}-10\pi x) \cdot (-10\pi) = -2\cos(\frac{\pi}{8}-10\pi x)$.
Ответ: $u'(x) = -2\cos(\frac{\pi}{8}-10\pi x)$.

б)

$f(x)=-22\cos\frac{x}{11}$
$f'(x) = (-22\cos\frac{x}{11})' = -22 \cdot (-\sin\frac{x}{11}) \cdot (\frac{x}{11})' = 22\sin(\frac{x}{11}) \cdot \frac{1}{11} = 2\sin(\frac{x}{11})$.
Ответ: $f'(x) = 2\sin(\frac{x}{11})$.

$g(x) = \frac{1}{18}\cos 27x$
$g'(x) = (\frac{1}{18}\cos 27x)' = \frac{1}{18} \cdot (-\sin 27x) \cdot (27x)' = -\frac{1}{18}\sin(27x) \cdot 27 = -\frac{27}{18}\sin(27x) = -\frac{3}{2}\sin(27x)$.
Ответ: $g'(x) = -\frac{3}{2}\sin(27x)$.

$h(x)=-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7})$
$h'(x) = (-\frac{2}{9}\cos(\frac{3x}{2}-\frac{\pi}{7}))' = -\frac{2}{9} \cdot (-\sin(\frac{3x}{2}-\frac{\pi}{7})) \cdot (\frac{3x}{2}-\frac{\pi}{7})' = \frac{2}{9}\sin(\frac{3x}{2}-\frac{\pi}{7}) \cdot \frac{3}{2} = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.
Ответ: $h'(x) = \frac{1}{3}\sin(\frac{3x}{2}-\frac{\pi}{7})$.

$u(x)=-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16})$
$u'(x) = (-\frac{8}{9}\cos(\frac{\pi}{8}-\frac{27\pi x}{16}))' = -\frac{8}{9} \cdot (-\sin(\frac{\pi}{8}-\frac{27\pi x}{16})) \cdot (\frac{\pi}{8}-\frac{27\pi x}{16})' = \frac{8}{9}\sin(\frac{\pi}{8}-\frac{27\pi x}{16}) \cdot (-\frac{27\pi}{16}) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.
Ответ: $u'(x) = -\frac{3\pi}{2}\sin(\frac{\pi}{8}-\frac{27\pi x}{16})$.

в)

$f(x)=4\text{tg}\frac{x}{4}$
$f'(x) = (4\text{tg}\frac{x}{4})' = 4 \cdot \frac{1}{\cos^2(\frac{x}{4})} \cdot (\frac{x}{4})' = \frac{4}{\cos^2(\frac{x}{4})} \cdot \frac{1}{4} = \frac{1}{\cos^2(\frac{x}{4})}$.
Ответ: $f'(x) = \frac{1}{\cos^2(\frac{x}{4})}$.

$g(x) = \frac{3}{4}\text{tg}\frac{8x}{9}$
$g'(x) = (\frac{3}{4}\text{tg}\frac{8x}{9})' = \frac{3}{4} \cdot \frac{1}{\cos^2(\frac{8x}{9})} \cdot (\frac{8x}{9})' = \frac{3}{4\cos^2(\frac{8x}{9})} \cdot \frac{8}{9} = \frac{2}{3\cos^2(\frac{8x}{9})}$.
Ответ: $g'(x) = \frac{2}{3\cos^2(\frac{8x}{9})}$.

$h(x) = \frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9})$
$h'(x) = (\frac{4}{45}\text{tg}(\frac{15x}{2}-\frac{\pi}{9}))' = \frac{4}{45} \cdot \frac{1}{\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot (\frac{15x}{2}-\frac{\pi}{9})' = \frac{4}{45\cos^2(\frac{15x}{2}-\frac{\pi}{9})} \cdot \frac{15}{2} = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.
Ответ: $h'(x) = \frac{2}{3\cos^2(\frac{15x}{2}-\frac{\pi}{9})}$.

$u(x) = \frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x)$
$u'(x) = (\frac{1}{2\pi}\text{tg}(\frac{\pi}{5}-5\pi x))' = \frac{1}{2\pi} \cdot \frac{1}{\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (\frac{\pi}{5}-5\pi x)' = \frac{1}{2\pi\cos^2(\frac{\pi}{5}-5\pi x)} \cdot (-5\pi) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.
Ответ: $u'(x) = -\frac{5}{2\cos^2(\frac{\pi}{5}-5\pi x)}$.

г)

$f(x)=-\frac{1}{3}\text{ctg}\frac{x}{3}$
$f'(x) = (-\frac{1}{3}\text{ctg}\frac{x}{3})' = -\frac{1}{3} \cdot (-\frac{1}{\sin^2(\frac{x}{3})}) \cdot (\frac{x}{3})' = \frac{1}{3\sin^2(\frac{x}{3})} \cdot \frac{1}{3} = \frac{1}{9\sin^2(\frac{x}{3})}$.
Ответ: $f'(x) = \frac{1}{9\sin^2(\frac{x}{3})}$.

$g(x) = \frac{9}{11}\text{ctg}\frac{22x}{3}$
$g'(x) = (\frac{9}{11}\text{ctg}\frac{22x}{3})' = \frac{9}{11} \cdot (-\frac{1}{\sin^2(\frac{22x}{3})}) \cdot (\frac{22x}{3})' = -\frac{9}{11\sin^2(\frac{22x}{3})} \cdot \frac{22}{3} = -\frac{6}{\sin^2(\frac{22x}{3})}$.
Ответ: $g'(x) = -\frac{6}{\sin^2(\frac{22x}{3})}$.

$h(x) = \frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21})$
$h'(x) = (\frac{3}{5}\text{ctg}(\frac{10x}{9}-\frac{\pi}{21}))' = \frac{3}{5} \cdot (-\frac{1}{\sin^2(\frac{10x}{9}-\frac{\pi}{21})}) \cdot (\frac{10x}{9}-\frac{\pi}{21})' = -\frac{3}{5\sin^2(\frac{10x}{9}-\frac{\pi}{21})} \cdot \frac{10}{9} = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.
Ответ: $h'(x) = -\frac{2}{3\sin^2(\frac{10x}{9}-\frac{\pi}{21})}$.

$u(x) = -\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x)$
$u'(x) = (-\frac{1}{25\pi}\text{ctg}(\frac{\pi}{11}-75\pi x))' = -\frac{1}{25\pi} \cdot (-\frac{1}{\sin^2(\frac{\pi}{11}-75\pi x)}) \cdot (\frac{\pi}{11}-75\pi x)' = \frac{1}{25\pi\sin^2(\frac{\pi}{11}-75\pi x)} \cdot (-75\pi) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.
Ответ: $u'(x) = -\frac{3}{\sin^2(\frac{\pi}{11}-75\pi x)}$.