# What is the derivative of #y = log3 [(x+1/x-1)^ln3]#?

I will guess that this should be:

So we get:

Note if the original should have been

Then we get

and

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To find the derivative of ( y = \log_3 \left[ \left( \frac{x+1}{x-1} \right)^{\ln 3} \right] ), we can use the chain rule and the properties of logarithms. The derivative is:

[ \frac{dy}{dx} = \frac{1}{\ln 3} \cdot \frac{1}{\ln 3} \cdot \frac{x-1}{x+1} \cdot \left( \frac{x+1}{x-1} \right)^{\ln 3} \cdot \left( \frac{x-1}{x+1} \right) ]

[ = \frac{(x-1)^2}{(x+1)^2} \cdot \left( \frac{x+1}{x-1} \right)^{\ln 3} ]

[ = \frac{(x-1)^2}{(x+1)^2} \cdot 3^{\ln \left( \frac{x+1}{x-1} \right)} ]

[ = \frac{(x-1)^2}{(x+1)^2} \cdot \left( \frac{x+1}{x-1} \right)^{\ln 3} ]

[ = \frac{(x-1)^2}{(x+1)^2} \cdot \left( \frac{x+1}{x-1} \right)^{\ln 3} ]

[ = \frac{(x-1)^2 \cdot 3^{\ln \left( \frac{x+1}{x-1} \right)}}{(x+1)^2} ]

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