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

Jan 6, 2016

$f ' \left(x\right) = \frac{\sin x - x \cos x}{\sin} ^ 2 x$. For "how", see below.

#### Explanation:

Quotient rule: $\frac{d}{\mathrm{dx}} \left(\frac{u}{v}\right) = \frac{u ' v - u v '}{v} ^ 2$

In this function, we have $u = x$ so $u ' = 1$

and $v = \sin x$, so $v ' = \cos x$.

Apply the rule:

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

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

This can, of course, be written in other ways using algebra and trigonometry.