Answer
$$f'\left( x \right) = \frac{{{{\tan }^{10}}x}}{{{{\left( {5x + 3} \right)}^6}}}\left( {\frac{{10}}{{\sin x\cos x}} - \frac{{30}}{{5x + 3}}} \right)$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{{{{\tan }^{10}}x}}{{{{\left( {5x + 3} \right)}^6}}} \cr
& {\text{taking the natural logarithm of both sides of the equation}} \cr
& \ln \left( {f\left( x \right)} \right) = \ln \left( {\frac{{{{\tan }^{10}}x}}{{{{\left( {5x + 3} \right)}^6}}}} \right) \cr
& {\text{quotient rule for logarithms}} \cr
& \ln \left( {f\left( x \right)} \right) = \ln \left( {{{\tan }^{10}}x} \right) - \ln \left( {{{\left( {5x + 3} \right)}^6}} \right) \cr
& {\text{power property for logarithms}} \cr
& \ln \left( {f\left( x \right)} \right) = 10\ln \left( {\tan x} \right) - 6\ln \left( {5x + 3} \right) \cr
& {\text{trigonometric identity for }}\tan x \cr
& \ln \left( {f\left( x \right)} \right) = 10\ln \left( {\frac{{\sin x}}{{\cos x}}} \right) - 6\ln \left( {5x + 3} \right) \cr
& \ln \left( {f\left( x \right)} \right) = 10\ln \left( {\sin x} \right) - 10\ln \left( {\cos x} \right) - 6\ln \left( {5x + 3} \right) \cr
& {\text{differentiate both sides with respect to }}x \cr
& \frac{{f'\left( x \right)}}{{f\left( x \right)}} = 10\left( {\frac{{\cos x}}{{\sin x}}} \right) - 10\left( {\frac{{ - \sin x}}{{\cos x}}} \right) - 6\left( {\frac{5}{{5x + 3}}} \right) \cr
& {\text{simplifying}} \cr
& \frac{{f'\left( x \right)}}{{f\left( x \right)}} = \frac{{10\cos x}}{{\sin x}} + \frac{{10\sin x}}{{\cos x}} - \frac{{30}}{{5x + 3}} \cr
& {\text{solving for }}f'\left( x \right) \cr
& f'\left( x \right) = f\left( x \right)\left( {\frac{{10{{\cos }^2}x + 10{{\sin }^2}x}}{{\sin x\cos x}} - \frac{{30}}{{5x + 3}}} \right) \cr
& f'\left( x \right) = f\left( x \right)\left( {\frac{{10}}{{\sin x\cos x}} - \frac{{30}}{{5x + 3}}} \right) \cr
& {\text{replace }}f\left( x \right){\text{ with the original function:}} \cr
& f'\left( x \right) = \frac{{{{\tan }^{10}}x}}{{{{\left( {5x + 3} \right)}^6}}}\left( {\frac{{10}}{{\sin x\cos x}} - \frac{{30}}{{5x + 3}}} \right) \cr} $$
