Differentiate the following functions with respect to x :

$\frac{e^{x} - t a n x}{c o t x - x^{n}}$

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Solution

Using quotient rule, we have,

$\frac{d}{d x} (\frac{e^{x} - t a n x}{c o t x - x^{n}})$

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

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

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

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