Подробное решение
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Не могу найти шаги в поиске этой производной.
Но производная
Ответ:
/ / /3 ___\\ /3 ___\ \
log(x)/3 ___\ |log\sin\\/ x // cos\\/ x /*log(x)|
sin \\/ x /*|--------------- + -----------------|
| x 2/3 /3 ___\|
\ 3*x *sin\\/ x //
$$\left(\frac{\log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{3 x^{\frac{2}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right) \sin^{\log{\left(x \right)}}{\left(\sqrt[3]{x} \right)}$$
/ 2 \
|/ / /3 ___\\ /3 ___\ \ |
||3*log\sin\\/ x // cos\\/ x /*log(x)| |
||----------------- + -----------------| |
|| x 2/3 /3 ___\ | / /3 ___\\ /3 ___\ /3 ___\ 2/3 ___\ |
log(x)/3 ___\ |\ x *sin\\/ x / / log\sin\\/ x // log(x) 2*cos\\/ x / 2*cos\\/ x /*log(x) cos \\/ x /*log(x)|
sin \\/ x /*|---------------------------------------- - --------------- - ------ + ----------------- - ------------------- - ------------------|
| 9 2 4/3 5/3 /3 ___\ 5/3 /3 ___\ 4/3 2/3 ___\|
\ x 9*x 3*x *sin\\/ x / 9*x *sin\\/ x / 9*x *sin \\/ x //
$$\left(\frac{\left(\frac{3 \log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{x^{\frac{2}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right)^{2}}{9} - \frac{\log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x^{2}} - \frac{\log{\left(x \right)}}{9 x^{\frac{4}{3}}} - \frac{\log{\left(x \right)} \cos^{2}{\left(\sqrt[3]{x} \right)}}{9 x^{\frac{4}{3}} \sin^{2}{\left(\sqrt[3]{x} \right)}} - \frac{2 \log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{9 x^{\frac{5}{3}} \sin{\left(\sqrt[3]{x} \right)}} + \frac{2 \cos{\left(\sqrt[3]{x} \right)}}{3 x^{\frac{5}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right) \sin^{\log{\left(x \right)}}{\left(\sqrt[3]{x} \right)}$$
/ 3 \
| / / /3 ___\\ /3 ___\ \ / / /3 ___\\ /3 ___\ \ / / /3 ___\\ /3 ___\ 2/3 ___\ /3 ___\ \ |
| |3*log\sin\\/ x // cos\\/ x /*log(x)| |3*log\sin\\/ x // cos\\/ x /*log(x)| |log(x) 9*log\sin\\/ x // 6*cos\\/ x / cos \\/ x /*log(x) 2*cos\\/ x /*log(x)| |
| |----------------- + -----------------| |----------------- + -----------------|*|------ + ----------------- - --------------- + ------------------ + -------------------| |
| | x 2/3 /3 ___\ | / /3 ___\\ | x 2/3 /3 ___\ | | 4/3 2 5/3 /3 ___\ 4/3 2/3 ___\ 5/3 /3 ___\ | /3 ___\ 2/3 ___\ 2/3 ___\ 3/3 ___\ /3 ___\ /3 ___\ |
log(x)/3 ___\ | 1 \ x *sin\\/ x / / 2*log\sin\\/ x // \ x *sin\\/ x / / \ x x x *sin\\/ x / x *sin \\/ x / x *sin\\/ x / / 2*log(x) 5*cos\\/ x / cos \\/ x / 2*cos \\/ x /*log(x) 2*cos \\/ x /*log(x) 2*cos\\/ x /*log(x) 10*cos\\/ x /*log(x)|
sin \\/ x /*|- ------ + ---------------------------------------- + ----------------- - --------------------------------------------------------------------------------------------------------------------------------- + -------- - ----------------- - ------------------ + -------------------- + -------------------- + ------------------- + --------------------|
| 7/3 27 3 9 7/3 8/3 /3 ___\ 7/3 2/3 ___\ 7/3 2/3 ___\ 2 3/3 ___\ 2 /3 ___\ 8/3 /3 ___\ |
\ 3*x x 9*x 3*x *sin\\/ x / 3*x *sin \\/ x / 9*x *sin \\/ x / 27*x *sin \\/ x / 27*x *sin\\/ x / 27*x *sin\\/ x / /
$$\left(\frac{\left(\frac{3 \log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{x^{\frac{2}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right)^{3}}{27} - \frac{\left(\frac{3 \log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x} + \frac{\log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{x^{\frac{2}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right) \left(\frac{9 \log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x^{2}} + \frac{\log{\left(x \right)}}{x^{\frac{4}{3}}} + \frac{\log{\left(x \right)} \cos^{2}{\left(\sqrt[3]{x} \right)}}{x^{\frac{4}{3}} \sin^{2}{\left(\sqrt[3]{x} \right)}} + \frac{2 \log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{x^{\frac{5}{3}} \sin{\left(\sqrt[3]{x} \right)}} - \frac{6 \cos{\left(\sqrt[3]{x} \right)}}{x^{\frac{5}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right)}{9} + \frac{2 \log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{27 x^{2} \sin{\left(\sqrt[3]{x} \right)}} + \frac{2 \log{\left(x \right)} \cos^{3}{\left(\sqrt[3]{x} \right)}}{27 x^{2} \sin^{3}{\left(\sqrt[3]{x} \right)}} + \frac{2 \log{\left(\sin{\left(\sqrt[3]{x} \right)} \right)}}{x^{3}} + \frac{2 \log{\left(x \right)}}{9 x^{\frac{7}{3}}} + \frac{2 \log{\left(x \right)} \cos^{2}{\left(\sqrt[3]{x} \right)}}{9 x^{\frac{7}{3}} \sin^{2}{\left(\sqrt[3]{x} \right)}} - \frac{1}{3 x^{\frac{7}{3}}} - \frac{\cos^{2}{\left(\sqrt[3]{x} \right)}}{3 x^{\frac{7}{3}} \sin^{2}{\left(\sqrt[3]{x} \right)}} + \frac{10 \log{\left(x \right)} \cos{\left(\sqrt[3]{x} \right)}}{27 x^{\frac{8}{3}} \sin{\left(\sqrt[3]{x} \right)}} - \frac{5 \cos{\left(\sqrt[3]{x} \right)}}{3 x^{\frac{8}{3}} \sin{\left(\sqrt[3]{x} \right)}}\right) \sin^{\log{\left(x \right)}}{\left(\sqrt[3]{x} \right)}$$