Differentiate the following functions with respect to x

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

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Solution

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

Differentiate both sides w.r.t. x,

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

[Using product rule $\frac{d}{d x} (u v) = u \frac{d}{d x} v + u \frac{d}{d x} v$ ]

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

[using chain rule $\frac{d}{d x} f (g (x)) = f^{'} (x) \frac{d}{d x} g (x)$ ]

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

(Using chain rule)

= $(c o s x^{3}) (2 s i n x^{5}) (c o s x^{5}) (5 x^{4}) - s i n^{2} (x^{5}) (s i n x^{3}) (3 x^{2})$

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