$(5$ marks$)$ Consider the function $f(x)=(sin(x)+cos(2x){)}^{ln(x)}.$ Find an expression for $f^{\text{'}}(x)$ and an equation for the tangent line at the point $(\frac{}{6},1).$

Given ${\displaystyle \underset{1}{\overset{3}{}}}f(x)dx=5,{\displaystyle \underset{3}{\overset{9}{}}}g(x)dx=4,{\displaystyle \underset{3}{\overset{9}{}}}h(x)dx=-6,$ and ${\displaystyle \underset{3}{\overset{1}{}}}f(x)dx={\displaystyle \underset{3}{\overset{9}{}}}\frac{4h(x)-g(x)}{2}dx={\displaystyle \underset{9}{\overset{1}{}}}r(x)dx=$