# How do you differentiate #f(x)=sqrt(1/csc(2/x ) # using the chain rule?

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To differentiate ( f(x) = \sqrt{1/\csc(2/x)} ) using the chain rule, follow these steps:

- Let ( u(x) = 1/\csc(2/x) ).
- Rewrite ( u(x) ) as ( u(x) = \sin(2/x) ).
- Let ( v(x) = \sqrt{u(x)} ).
- Apply the chain rule: ( f'(x) = v'(x) = \frac{dv}{du} \cdot \frac{du}{dx} ).
- Find ( \frac{dv}{du} ) and ( \frac{du}{dx} ).
- ( \frac{dv}{du} = \frac{1}{2\sqrt{u(x)}} ).
- ( \frac{du}{dx} = \frac{d}{dx}[\sin(2/x)] = \cos(2/x) \cdot \frac{d}{dx}(2/x) ).
- Use the chain rule again for ( \frac{d}{dx}(2/x) ): ( \frac{d}{dx}(2/x) = -\frac{2}{x^2} ).
- Combine the results: ( f'(x) = \frac{1}{2\sqrt{1/\csc(2/x)}} \cdot \cos(2/x) \cdot \left(-\frac{2}{x^2}\right) ).
- Simplify the expression for ( f'(x) ): ( f'(x) = -\frac{\cos(2/x)}{x^2\sqrt{\csc(2/x)}} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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