# How do you find the derivative of #f(x)=log_2(x^2/(x-1))#?

We need to know the logarithm rules:

We also have to know that:

Then:

The derivative is then:

Which can be simplified:

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To find the derivative of ( f(x) = \log_2\left(\frac{x^2}{x-1}\right) ), you can use the chain rule and the properties of logarithms. The derivative can be calculated as follows:

[ f'(x) = \frac{d}{dx}\left[\log_2\left(\frac{x^2}{x-1}\right)\right] ]

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

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

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

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

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

So, the derivative of ( f(x) ) is ( \frac{x^2 - 2x}{(x-1)^2 \cdot \ln(2)} ).

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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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