# Use the Quotient Rule to show that \frac{d}{dx}(\csc x) = - \csc(x) \cot(x)

## Question:

Use the Quotient Rule to show that

{eq}\frac{d}{dx}(\csc x) = - \csc(x) \cot(x) {/eq}

## Quotient Rule:

The quotient rule is a differentiation rule that is applied when we are differentiating functions that are expressed as a quotient of distinct functions. This rule considers the derivative of both the numerator and denominator of the function through the formula, {eq}\displaystyle d\left( \frac{u}{v}\right) = \frac{vdu-udv}{v^2} {/eq}.

The quotient rule is given as {eq}\displaystyle d\left( \frac{u}{v}\right) = \frac{vdu-udv}{v^2} {/eq}, and we apply this formula to differentiate {eq}\displaystyle \csc x = \frac{1}{\sin x} {/eq}, such that {eq}\displaystyle u = 1 {/eq} and {eq}\displaystyle v = \sin x {/eq}. We proceed with the solution.

{eq}\begin{align} \displaystyle \frac{d}{dx}\csc x &= \frac{d}{dx} \frac{1}{\sin x}\\ &= \frac{\sin x \cdot 0- 1 \cdot \cos x}{\sin^2 x}\\ &= -\frac{\cos x}{\sin ^2 x}\\ &= - \frac{1}{\sin x} \frac{\cos x}{\sin x}\\ &= -\csc x\cot x \end{align} {/eq}