# How do you differentiate f(x)=cscx using the quotient rule?

Dec 9, 2015

First we must write $f \left(x\right)$ as a quotient.

#### Explanation:

$f \left(x\right) = \csc x = \frac{1}{\sin} x$

Now use the quotient rule to differentiate:

$f ' \left(x\right) = \frac{\left(0\right) \left(\sin x\right) - \left(1\right) \left(\cos x\right)}{\sin x} ^ 2$

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

$= - \cos \frac{x}{\sin} x \frac{1}{\sin} x$

$= - \cot x \csc x = - \csc x \cot x$

(Choose a way of expressing the answer that you (or your grader) prefer.)