# How do you use the limit definition to find the derivative of y = cscx?

Dec 2, 2016

The key parts are in the explanation section, below.

#### Explanation:

$\frac{\frac{1}{\sin} \left(x + h\right) - \frac{1}{\sin} x}{h} = \frac{\sin x - \sin \left(x + h\right)}{h \sin \left(x + h\right) \sin x}$

$= \frac{\sin x - \sin x \cos h - \cos x \sin h}{h \sin \left(x + h\right) \sin x}$

$= \left(\sin x \frac{1 - \cos h}{h} - \cos x \sin \frac{h}{h}\right) \cdot \frac{1}{\sin \left(x + h\right) \sin x}$

Taking the limit gets us

$\frac{- \cos x}{\sin} ^ 2 x = - \csc x \cot x$