Differentiate the following functions with respect to x :

$\frac{x + c o s x}{t a n x}$

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Solution

Let $u = x + c o s x, v = t a n x$

Then $u^{'} = 1 - s i n x, v^{'} = s e c^{2} x$

Using the quotient rule

$\frac{d y}{d x} (\frac{u}{v}) = \frac{v u^{'} - u v^{'}}{v^{2}}$

$\frac{d}{d x} (\frac{x + c o s x}{t a n x}) = \frac{t a n x (1 - s i n x) - (x + c o s x) s e c^{2} x}{t a n^{2} x}$

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