Question

Differentiate the function w.r.t.x:
$\rm \frac{x^{2}-\cos x}{\sin x}$

Answer 1

We know that such form of expression can solve by U/V method of differentiation

$\frac{d}{dx} \bigg(\frac{U}{V}\bigg ) = \frac{V \frac{dU}{dx} - U \frac{dV}{dx} }{ {V}^{2} } \\ \\ here \: U = x^{2}-cos\:x \\ \\ V = sin\:x$

$\frac{d}{dx} \bigg(\frac{ {x}^{2} - cos \: x}{sin \: x}\bigg ) = \frac{sin \: x \frac{d( {x}^{2} - cos \: x) }{dx} - ({x}^{2} - cos \: x) \frac{d \: sin \: x}{dx} }{ {(sin \: x})^{2} } \\ \\ = \frac{sin \: x (2x + sin \:x)- cos \: x({x}^{2} - cos \: x) }{ {sin}^{2}x } \\ \\ = \frac{2x \: sin \: x + {sin}^{2}x - {x}^{2} cos \: x + {cos}^{2}x }{ {sin}^{2}x } \\ \\ \frac{d}{dx} \bigg(\frac{ {x}^{2} - cos \: x}{sin \: x}\bigg ) = \frac{2x \: sin \: x - {x}^{2} cos \: x + 1}{ {sin}^{2x} }$
Hope it helps you.