# if

Question:

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

Solution:

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

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

$\frac{1}{y} \frac{d y}{d x}=3\left[\log 5 x \cdot \frac{d}{d x}(\cos 2 x)+\cos 2 x \cdot \frac{d}{d x}(\log 5 x)\right]$

$\Rightarrow \frac{d y}{d x}=3 y\left[\log 5 x(-\sin 2 x) \cdot \frac{d}{d x}(2 x)+\cos 2 x \cdot \frac{1}{5 x} \cdot \frac{d}{d x}(5 x)\right]$

$\Rightarrow \frac{d y}{d x}=3 y\left[-2 \sin 2 x \log 5 x+\frac{\cos 2 x}{x}\right]$

$\Rightarrow \frac{d y}{d x}=3 y\left[\frac{3 \cos 2 x}{x}-6 \sin 2 x \log 5 x\right]$

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