Solución detallada
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No logro encontrar los pasos en la búsqueda de esta derivada.
Perola derivada
Respuesta:
sin(x)
/ ___\ / / ___\\
| \/ x | |/ 1 log(x)\ | \/ x ||
\x / *||----- + -------|*sin(x) + cos(x)*log\x /|
|| ___ ___| |
\\\/ x 2*\/ x / /
$$\left(\left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right) \sin{\left(x \right)} + \log{\left(x^{\sqrt{x}} \right)} \cos{\left(x \right)}\right) \left(x^{\sqrt{x}}\right)^{\sin{\left(x \right)}}$$
/ 2 \
|/ / ___\ \ |
|| | \/ x | (2 + log(x))*sin(x)| |
sin(x) ||2*cos(x)*log\x / + -------------------| |
/ ___\ || ___ | / ___\ |
| \/ x | |\ \/ x / | \/ x | (2 + log(x))*cos(x) log(x)*sin(x)|
\x / *|--------------------------------------------- - log\x /*sin(x) + ------------------- - -------------|
| 4 ___ 3/2 |
\ \/ x 4*x /
$$\left(\frac{\left(2 \log{\left(x^{\sqrt{x}} \right)} \cos{\left(x \right)} + \frac{\left(\log{\left(x \right)} + 2\right) \sin{\left(x \right)}}{\sqrt{x}}\right)^{2}}{4} - \log{\left(x^{\sqrt{x}} \right)} \sin{\left(x \right)} + \frac{\left(\log{\left(x \right)} + 2\right) \cos{\left(x \right)}}{\sqrt{x}} - \frac{\log{\left(x \right)} \sin{\left(x \right)}}{4 x^{\frac{3}{2}}}\right) \left(x^{\sqrt{x}}\right)^{\sin{\left(x \right)}}$$
/ 3 \
|/ / ___\ \ / / ___\ \ / / ___\ \ |
|| | \/ x | (2 + log(x))*sin(x)| | | \/ x | (2 + log(x))*sin(x)| | | \/ x | log(x)*sin(x) 4*(2 + log(x))*cos(x)| |
sin(x) ||2*cos(x)*log\x / + -------------------| 3*|2*cos(x)*log\x / + -------------------|*|4*log\x /*sin(x) + ------------- - ---------------------| |
/ ___\ || ___ | / ___\ | ___ | | 3/2 ___ | |
| \/ x | |\ \/ x / | \/ x | \ \/ x / \ x \/ x / sin(x) 3*(2 + log(x))*sin(x) 3*cos(x)*log(x) 3*log(x)*sin(x)|
\x / *|--------------------------------------------- - cos(x)*log\x / - ------------------------------------------------------------------------------------------------------------- - ------ - --------------------- - --------------- + ---------------|
| 8 8 5/2 ___ 3/2 5/2 |
\ 4*x 2*\/ x 4*x 8*x /
$$\left(\frac{\left(2 \log{\left(x^{\sqrt{x}} \right)} \cos{\left(x \right)} + \frac{\left(\log{\left(x \right)} + 2\right) \sin{\left(x \right)}}{\sqrt{x}}\right)^{3}}{8} - \frac{3 \left(2 \log{\left(x^{\sqrt{x}} \right)} \cos{\left(x \right)} + \frac{\left(\log{\left(x \right)} + 2\right) \sin{\left(x \right)}}{\sqrt{x}}\right) \left(4 \log{\left(x^{\sqrt{x}} \right)} \sin{\left(x \right)} - \frac{4 \left(\log{\left(x \right)} + 2\right) \cos{\left(x \right)}}{\sqrt{x}} + \frac{\log{\left(x \right)} \sin{\left(x \right)}}{x^{\frac{3}{2}}}\right)}{8} - \log{\left(x^{\sqrt{x}} \right)} \cos{\left(x \right)} - \frac{3 \left(\log{\left(x \right)} + 2\right) \sin{\left(x \right)}}{2 \sqrt{x}} - \frac{3 \log{\left(x \right)} \cos{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \log{\left(x \right)} \sin{\left(x \right)}}{8 x^{\frac{5}{2}}} - \frac{\sin{\left(x \right)}}{4 x^{\frac{5}{2}}}\right) \left(x^{\sqrt{x}}\right)^{\sin{\left(x \right)}}$$