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

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

- Identify the outer function and the inner function. In this case, the outer function is ( \csc(x) ) and the inner function is ( \ln(1/x) ).
- Take the derivative of the outer function with respect to the inner function, and then multiply by the derivative of the inner function with respect to ( x ).
- The derivative of ( \csc(x) ) is ( -\csc(x) \cot(x) ).
- The derivative of ( \ln(1/x) ) with respect to ( x ) is ( -1/x ).
- Combine the results from steps 3 and 4 to get the final derivative.

Therefore, the derivative of ( f(x) = \csc(\ln(1/x)) ) with respect to ( x ) is:

[ f'(x) = -\csc(\ln(1/x)) \cot(\ln(1/x)) \times \left(-\frac{1}{x}\right) ]

[ f'(x) = \frac{\csc(\ln(1/x)) \cot(\ln(1/x))}{x} ]

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