The quotient rule is worth knowing, but it is not always worth using. f/g is f·g⁻¹, so the
product rule and the chain rule between them can do the same job — often with less algebra and fewer
sign errors.
When the quotient rule wins
When the denominator is a genuine expression that will stay in the answer — (x²+1)/(x−1) — the
quotient rule’s single fraction is tidier than expanding a negative power.
When it loses
When the denominator is a constant, it is not a quotient at all: x²/5 is ⅕x², and the constant
multiple rule finishes it in one line. When the denominator is a single power, 3/x² is 3x⁻², and the
power rule is quicker than a two-term numerator.
The sign
If you do use it, the order matters: low times d-high, minus high times d-low. Reversing it flips the sign of the entire answer, and because the denominator is squared, nothing downstream will warn you.