${(5x)^{3\cos 2x}}$

Let $y =$ ${(5x)^{3\cos 2x}}$

By taking log on both sides , we get

$\log y = 3\cos 2x\log (5x) = 3\cos 2x[\log 5 + \log x]$

$\log y = 3\cos 2x\log 5 + 3\cos 2x\log x$ …(i)

Differentiating (i) w.r.t. x, we get

$\cfrac{1}{y}\cfrac{{dy}}{{dx}} = 3\log 5( - \sin 2x) \cdot 2 + \cfrac{{3\cos 2x}}{x} - 3\log x \cdot (2 \cdot \sin 2x)$

$= - 6\log 5\sin 2x + \cfrac{{3\cos 2x}}{x} - 6\log x\sin 2x$

therefore,$\cfrac{{dy}}{{dx}} = {(5x)^{3\cos 2x}}\left[ {\cfrac{{3\cos 2x}}{x} - 6[\log 5 + \log x]\sin 2x} \right]$

$= {(5x)^{3\cos 2x}}\left[ {\cfrac{{3\cos 2x}}{x} - 6\log 5x\sin 2x} \right]$