[tex]y=(\sin(7x))^{\ln(x) }
\\
\\ \ln{y}=\ln[(\sin(7x))^{\ln(x) }]
\\
\\ \ln{y}=\ln(x)\ln[(\sin(7x))]
\\
\\ (\ln{y})'=(\ln(x)\ln[(\sin(7x))])'
\\
\\ \frac{1}{y} dy=[\frac{1}x}\ ln \sin(7x) +\frac{7 \cos (7x)}{sin(7x)}\ln{x}] dx
\\
\\ \frac{dy}{dx} =y[\frac{1}x}\ ln \sin(7x) +\frac{7 \cos (7x)}{sin(7x)}\ln{x}]
\\
\\ \frac{dy}{dx} =(\sin(7x))^{\ln(x) }[\frac{1}x}\ ln \sin(7x) +\frac{7 \cos (7x)}{sin(7x)}\ln{x}] [/tex]
