Quotient Rule
The quotient rule gives the derivative of one function divided by another: $(f/g)' = \dfrac{f'g - fg'}{g^2}$.
Formula
\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}
Definition
The quotient rule finds the derivative of a fraction where both top and bottom are functions, following the mnemonic "low d-high minus high d-low, square the bottom and away we go." If $g(x) \neq 0$, then $d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2$, which can be derived from the product rule applied to $f(x) \cdot [g(x)]^{-1}$, making it a direct consequence of the product rule and chain rule. For holomorphic (complex) functions it holds in exactly the same form.
Example
Differentiating $x^2 / \sin(x)$ gives $(2x\sin x - x^2\cos x) / \sin^2(x)$. For $f(x) = (x^2 + 1)/(x - 2)$: $f'(x) = [2x(x-2) - (x^2+1)(1)] / (x-2)^2 = (x^2 - 4x - 1)/(x-2)^2$. Differentiating $\tan(x) = \sin(x)/\cos(x)$ via the quotient rule gives $(\cos^2 x + \sin^2 x)/\cos^2 x = 1/\cos^2 x = \sec^2 x$, a standard result used throughout physics.
Key Insight
The quotient rule always subtracts (not adds) in the numerator, so order matters, switching $f$ and $g$ gives a different answer; many teachers prefer rewriting division as multiplication by a negative power and using the product/chain rules instead, which avoids the subtraction-order trap. In algebraic geometry, the quotient rule generalizes to the derivation of function fields, connecting differential calculus to the study of algebraic varieties.