Differentiate given problems w.r.t.x.

$(5 x)^{3 c o s 2 x}$

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Solution

Let y = $(5 x)^{3 c o s 2 x}$

Taking log on both sides, we get log y = log $(5 x)^{3 c o s 2 x}$ =3 cos 2x (log 5x) Differentiating w.r.t x. we get

$\frac{1}{y} \frac{d y}{d x} = 3 c o s 2 x \frac{d}{d x} (l o g 5 x) \frac{d}{d x} (3 c o s 2 x)$

= $(3 c o s 2 x) (\frac{5}{5 x}, 5) + (l o g 5 x) [3 (- s i n 2 x) .2]$

$(U s i n g p r o d u c t r u l e \frac{d}{d x} (u . v) = u \frac{d}{d x} v + v \frac{d}{d x} u)$

$\frac{d y}{d x} = y {\frac{3 c o s 2 x}{x} - 6 (s i n 2 x) (l o g 5 x)}$

$\frac{d y}{d x} = (5 x)^{3 c o s 2 x} {\frac{3 c o s 2 x}{x} - 6 s i n 2 x l o g 5 x}$

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