How do you differentiate #f(x)=sqrt(csc(2/x ) # using the chain rule?

First take the derivative of sqrt(exponent of 1/2) while keeping everything else inside the same next take the derivative of csc while keeping (2/x) the same and finally the derivative of (2/x)

To differentiate ( f(x) = \sqrt{\csc\left(\frac{2}{x}\right)} ) using the chain rule, we follow these steps:

1. Let ( u = \frac{2}{x} ).
2. Find ( \frac{du}{dx} ).
3. Find ( \frac{df}{du} ).
4. Multiply ( \frac{df}{du} ) by ( \frac{du}{dx} ) to get ( \frac{df}{dx} ).

( \frac{du}{dx} = -\frac{2}{x^2} )

( \frac{df}{du} = \frac{1}{2\sqrt{\csc(u)}}\csc(u)\cot(u) )

( \frac{df}{dx} = \frac{df}{du} \times \frac{du}{dx} = \frac{1}{2\sqrt{\csc(u)}}\csc(u)\cot(u) \times -\frac{2}{x^2} = -\frac{\csc(u)\cot(u)}{x^2\sqrt{\csc(u)}} )

Replace ( u ) with ( \frac{2}{x} ) to get the final answer:

( \frac{df}{dx} = -\frac{\csc\left(\frac{2}{x}\right)\cot\left(\frac{2}{x}\right)}{x^2\sqrt{\csc\left(\frac{2}{x}\right)}} )