Differentiate the following functions with respect to x:
$(x s i n x + c o s x) (x c o s x - s i n x)$

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Solution

We have,

$\frac{d}{d x} (x s i n x + c o s x) (x c o s x - s i n x)$

We will apply product rule,

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

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

$= x^{n - 1} [n . l o g x_{a} + \frac{1}{l o g a}]$

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