# How do you find the derivative of #ln(1+1/x) / (1/x)#?

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To find the derivative of ln(1+1/x) / (1/x), we can use the quotient rule. Let u = ln(1+1/x) and v = (1/x). Then, using the quotient rule (u/v)' = (u'v - uv') / v^2, we find that the derivative is:

[(((1 - (1/x)) * (-1/x^2)) * (1/x)) - (ln(1+1/x) * (-1/x^2))] / (1/x)^2

Simplifying this expression yields the derivative:

= [((-1 + (1/x)) * (1/x^2)) + (ln(1+1/x) * (1/x^2))] / (1/x)^2

= [((1/x) * (1/x^2)) + (ln(1+1/x) * (1/x^2))] / (1/x)^2

= [(1/x^3) + (ln(1+1/x) / x^2)] / (1/x^2)

= x * (1/x^3) + (ln(1+1/x) / x^2)

= 1/x^2 + (ln(1+1/x) / x^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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