Solución detallada
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No logro encontrar los pasos en la búsqueda de esta derivada.
Perola derivada
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Simplificamos:
Respuesta:
/ / /x\\ /x\\
|4*log|cos|-|| log(x + 3*x)*sin|-||
log(x + 3*x)/x\ | \ \4// \4/|
cos |-|*|------------- - -------------------|
\4/ | x + 3*x /x\ |
| 4*cos|-| |
\ \4/ /
$$\left(- \frac{\log{\left(x + 3 x \right)} \sin{\left(\frac{x}{4} \right)}}{4 \cos{\left(\frac{x}{4} \right)}} + \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x + 3 x}\right) \cos^{\log{\left(x + 3 x \right)}}{\left(\frac{x}{4} \right)}$$
/ 2 \
| / / /x\\ /x\\ |
| | 4*log|cos|-|| log(4*x)*sin|-|| |
| | \ \4// \4/| |
| |- ------------- + ---------------| |
| | x /x\ | / /x\\ /x\ 2/x\ |
| | cos|-| | log|cos|-|| sin|-| sin |-|*log(4*x)|
log(4*x)/x\ | log(4*x) \ \4/ / \ \4// \4/ \4/ |
cos |-|*|- -------- + ------------------------------------ - ----------- - ---------- - ----------------|
\4/ | 16 16 2 /x\ 2/x\ |
| x 2*x*cos|-| 16*cos |-| |
\ \4/ \4/ /
$$\left(\frac{\left(\frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{\cos{\left(\frac{x}{4} \right)}} - \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x}\right)^{2}}{16} - \frac{\log{\left(4 x \right)} \sin^{2}{\left(\frac{x}{4} \right)}}{16 \cos^{2}{\left(\frac{x}{4} \right)}} - \frac{\log{\left(4 x \right)}}{16} - \frac{\sin{\left(\frac{x}{4} \right)}}{2 x \cos{\left(\frac{x}{4} \right)}} - \frac{\log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x^{2}}\right) \cos^{\log{\left(4 x \right)}}{\left(\frac{x}{4} \right)}$$
/ 3 \
| / / /x\\ /x\\ / / /x\\ /x\\ / / /x\\ 2/x\ /x\ \ |
| | 4*log|cos|-|| log(4*x)*sin|-|| | 4*log|cos|-|| log(4*x)*sin|-|| |16*log|cos|-|| sin |-|*log(4*x) 8*sin|-| | |
| | \ \4// \4/| | \ \4// \4/| | \ \4// \4/ \4/ | |
| |- ------------- + ---------------| 3*|- ------------- + ---------------|*|-------------- + ---------------- + -------- + log(4*x)| |
| | x /x\ | / /x\\ | x /x\ | | 2 2/x\ /x\ | 2/x\ /x\ 3/x\ /x\ |
| | cos|-| | 2*log|cos|-|| | cos|-| | | x cos |-| x*cos|-| | 3*sin |-| log(4*x)*sin|-| sin |-|*log(4*x) 3*sin|-| |
log(4*x)/x\ | 3 \ \4/ / \ \4// \ \4/ / \ \4/ \4/ / \4/ \4/ \4/ \4/ |
cos |-|*|- ---- - ------------------------------------ + ------------- + ----------------------------------------------------------------------------------------------- - ------------ - --------------- - ---------------- + -----------|
\4/ | 16*x 64 3 64 2/x\ /x\ 3/x\ 2 /x\|
| x 16*x*cos |-| 32*cos|-| 32*cos |-| 4*x *cos|-||
\ \4/ \4/ \4/ \4//
$$\left(- \frac{\left(\frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{\cos{\left(\frac{x}{4} \right)}} - \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x}\right)^{3}}{64} + \frac{3 \left(\frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{\cos{\left(\frac{x}{4} \right)}} - \frac{4 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x}\right) \left(\frac{\log{\left(4 x \right)} \sin^{2}{\left(\frac{x}{4} \right)}}{\cos^{2}{\left(\frac{x}{4} \right)}} + \log{\left(4 x \right)} + \frac{8 \sin{\left(\frac{x}{4} \right)}}{x \cos{\left(\frac{x}{4} \right)}} + \frac{16 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x^{2}}\right)}{64} - \frac{\log{\left(4 x \right)} \sin^{3}{\left(\frac{x}{4} \right)}}{32 \cos^{3}{\left(\frac{x}{4} \right)}} - \frac{\log{\left(4 x \right)} \sin{\left(\frac{x}{4} \right)}}{32 \cos{\left(\frac{x}{4} \right)}} - \frac{3 \sin^{2}{\left(\frac{x}{4} \right)}}{16 x \cos^{2}{\left(\frac{x}{4} \right)}} - \frac{3}{16 x} + \frac{3 \sin{\left(\frac{x}{4} \right)}}{4 x^{2} \cos{\left(\frac{x}{4} \right)}} + \frac{2 \log{\left(\cos{\left(\frac{x}{4} \right)} \right)}}{x^{3}}\right) \cos^{\log{\left(4 x \right)}}{\left(\frac{x}{4} \right)}$$