Use logarithmic differentiation to find the derivative of the function $f\left(x\right)=\frac{x^5{\sin }^{2}x}{\sqrt{x+7^{}}}$.

$f^{\prime }\left(x\right)=-\frac{2\sqrt{x+7}}{x^5{\sin }^{2}x}$

$f^{\prime }\left(x\right)=\frac{x^5{\sin }^{2}x(3x^2\tan x+4x-9)}{2\sqrt{x+7}}$

$f^{\prime }\left(x\right)=\frac{x^4{\sin }^{2}x(4x^2\cot x+28x\cot x+9x+70)}{2{(x+7)}^{\frac{3}{2}}}$

$f^{\prime }\left(x\right)=\frac{2\sqrt{x+7}}{x^5{\sin }^{2}x}$