# How do you find the first and second derivative of #(ln(x^2-1))/x^2#?

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To find the first and second derivative of (\frac{{\ln(x^2-1)}}{{x^2}}), use the quotient rule for differentiation.

Let (u = \ln(x^2-1)) and (v = x^{-2}). Then, the first derivative is:

[ \begin{aligned} \frac{d}{dx}\left(\frac{u}{v}\right) &= \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} \ &= \frac{x^{-2}\frac{d}{dx}(\ln(x^2-1)) - \ln(x^2-1)\frac{d}{dx}(x^{-2})}{x^{-4}}. \end{aligned} ]

Now, find (\frac{d}{dx}(\ln(x^2-1))) and (\frac{d}{dx}(x^{-2})) using the chain rule and the power rule, respectively. Then, substitute these into the formula above to find the first derivative. Once you have the first derivative, you can find the second derivative by differentiating the first derivative.

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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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