# Find the derivative of the function f(x)=e^[x/(x-1)].

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We have to find the derivative of the function f(x) = e^(x/(x-1).

Let's use the chain rule:

f'(x) = e^(x/(x-1)*(x/(x-1)'

=> e^(x/(x-1)*[(x-1) - (x))/(x - 1)^2]

=> e^(x/(x-1)*[-1/(x - 1)^2]

**The required derivative is e^(x/(x-1)*[(- 1)/(x - 1)^2]**

We'll apply chain rule to determine the derivative of the function:

f'(x) = e^[x/(x-1)]*[x/(x-1)]'

Since we have to differentiate a fraction, we'll apply quotient rule:

(u/v)' = (u'v - uv')/v^2

u = x => u' = 1

v = (x-1) => v' = 1

[x/(x-1)]' = [(x-1) - x]/(x-1)^2

We'll eliminate like terms:

[x/(x-1)]' = -1/(x-1)^2

The derivative of the function is:

**f'(x) = [-1/(x-1)^2]*e^[x/(x-1)]**