Differentiate the given function w.r.t. $x$ :
$(5 x)^{3 cos 2 x}$

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Solution

We have $y = (5 x)^{3 cos 2 x}$
Taking logarithm on both the sides, we obtain
$log y = 3 cos 2 x log 5 x$
Differentiating both sides with respect to $x$ , we obtain
$\frac{1}{y} \frac{d y}{d x} = 3 [log 5 x \frac{d}{d x} (cos 2 x) + cos 2 x \frac{d}{d x} (log 5 x)]$
$\Rightarrow \frac{d}{d x} = 3 y [log 5 x (- sin 2 x) . \frac{d}{d x} (2 x) + cos 2 x . \frac{1}{5 x} \frac{d}{d x} (5 x)]$
$\Rightarrow \frac{d y}{d x} = 3 x [- 2 sin 2 x log 5 x + \frac{cos 2 x}{x}]$
$\Rightarrow \frac{d y}{d x} = 3 y [\frac{3 cos 2 x}{x} - 6 sin 2 x log 5 x]$
$∴ \frac{d y}{d x} = (5 x)^{3 cos 2 x} [\frac{3 cos 2 x}{x} - 6 sin 2 x log 5 x]$

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