Find the derivative of $s i n (x^{2}) + (s i n x)^{2} + s i n^{2} (x^{2})$

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Solution

Let $y = s i n (x^{2}) + (s i n x)^{2} + s i n^{2} (x^{2}) \frac{d y}{d x} = \frac{d}{d x} s i n (x^{2}) + \frac{d}{d x} (s i n x)^{2} + \frac{d}{d x} (s i n x^{2})^{2} = c o s (x^{2}) \frac{d}{d x} (x^{2}) + 2 s i n x . \frac{d}{d x} + 2 s i n x^{2} . \frac{d}{d x} s i n x^{2} = c o s x^{2} 2 x + x . s o m x . c o s x + 2 s i n x^{2} c o s x^{2} . \frac{d}{d x} x^{2} = 2 x c o s (x)^{2} + 2. s i n x . c o s x + 2 s i n x^{2} . c o s x^{2} .2 x = 2 x c o s (x^{2}) + s i n 2 x + s i n 2 (x)^{2} .2 x = 2 x c o s (x^{2}) + 2 x . s i n 2 (x^{2}) + s i n 2 x$

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