I have the following the function: f(x) = \dfrac{x}{x+1}. Should I use the quotient rule to differentiate it, or should I use the product rule with f(x) = x(x+1)^{-1}

You can use either one! It honestly is just personal preference. Personally, I tend to mostly use the product rule because it’s easier to remember than the quotient rule. But the quotient rule is faster to use, if you can remember the rule.

Quotient Rule: \dfrac{(1)(x+1) - x(1)}{(x+1)^2} = \dfrac{1}{(x+1)^2} \tag*{}

Product Rule: \begin{align} (1)(x+1)^{-1} + (x)(-1)(x+1)^{-2} &= \dfrac{(1)(x+1) - x(1)}{(x+1)^2} \\ &= \dfrac{1}{(x+1)^2} \end{align} \tag*{}

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