# How do you differentiate f(x)= ( x + 1 )/ ( cot 3x ) using the quotient rule?

##### 1 Answer
Feb 11, 2018

$f ' \left(x\right) = \frac{\cot \left(3 x\right) + 3 \left(x + 1\right) {\csc}^{2} \left(3 x\right)}{{\cot}^{2} \left(3 x\right)}$

#### Explanation:

$\text{given "f(x)=(g(x))/(h(x))" then}$

$f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2 \leftarrow \textcolor{b l u e}{\text{quotient rule}}$

$g \left(x\right) = x + 1 \Rightarrow g ' \left(x\right) = 1$

$h \left(x\right) = \cot \left(3 x\right) \Rightarrow h ' \left(x\right) = - 3 {\csc}^{2} \left(3 x\right) \leftarrow \textcolor{b l u e}{\text{chain rule}}$

$f ' \left(x\right) = \frac{\cot \left(3 x\right) + 3 \left(x + 1\right) {\csc}^{2} \left(3 x\right)}{{\cot}^{2} \left(3 x\right)}$