Question

find the derivative by using first principle x upon sin x​

Answer 1

$\\ \frac{d}{dx} ( \frac{x}{ \sin(x) } )$

$\\ = \lim_{h \to0} (\frac{ \frac{x + h}{ \sin(x + h) } - \frac{x}{ \sin(x) } }{h} )$

$\\ = \lim_{h \to0}( \frac{(x + h) \sin(x) - x \sin(x + h) }{h \sin(x) \sin(x + h) } )$

$\\ = \lim_{h \to0} \frac{x( \sin(x) - \sin(x + h) ) + h \sin(x) }{h \sin(x) \sin(x + h) }$

$\\ \small{ =\frac{1}{ \sin(x) } + x \lim_{h \to0} \frac{( \sin(x) - \sin(x + h)) }{h \sin^{2} (x) } }$

$\\ \small{ = \cosec(x) + \frac{x}{ \sin(x) } \lim_{h \to0} \frac{ 1 - \cos(h) }{h } - \frac{x \cos(x) }{ \sin^{2} (x) } \lim_{h \to0} \frac{ \sin(h) }{h} }$

$= \small{ \cosec(x) - x \cot(x) \cosec(x) }$

$= \frac{ \sin(x) - x \cos(x) }{ \sin^{2} (x) }$