Differentiation of a quotient

Let $y = \frac{u}{v} v \ne 0$. By the General Rule,

• FIRST STEP. $y + \Delta y = \frac{u + \Delta u}{v + \Delta v}$.
• SECOND STEP. $\Delta y = \frac{u + \Delta u}{v \Delta v} - \frac{u}{v} = \frac{v \cdot \Delta u - u \cdot \Delta v}{v(v + \Delta v)}$.
• THIRD STEP. $\frac{\Delta y}{\Delta x} = \frac{v\frac{\Delta u}{\Delta x} - u\frac{\Delta v}{\Delta x}}{v(v + \Delta v)}$.
• FOURTH STEP. $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. [Applying Theorems 3.8.2 and 3.8.3]

Therefore, $\frac{d}{dx} \left ( \frac{u}{v} \right ) \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ (equation (VII) above).

The derivative of a fraction is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

When the denominator is constant, set $v = c$ in (VII), giving (VIIa) $\frac{d}{dx} \left ( \frac{u}{c} \right ) = \frac{\frac{du}{dx}}{c}$. [Since $\frac{dv}{dx} = \frac{dc}{dx} = 0$.] We may also get (VIIa) from (IV) as follows:

$$\frac{d}{dx} \left ( \frac{u}{c} \right ) = \frac{1}{c} \frac{du}{dx} = \frac{\frac{du}{dx}}{c}.$$

The derivative of the quotient of a function by a constant is equal to the derivative of the function divided by the constant.

All explicit algebraic functions of one independent variable may be differentiated by following the rules we have deduced so far.