Quotient Rule

104. Quotient Rule

\frac{𝑑}{𝑑 𝑥} [\frac{𝑓 (𝑥)}{𝑔 (𝑥)}] = \frac{𝑓^{'} (𝑥) 𝑔 (𝑥) - 𝑓 (𝑥) 𝑔^{'} (𝑥)}{{[𝑔 (𝑥)]}^{2}}

Example

\frac{𝑑}{𝑑 𝑥} [\frac{𝑥^{2}}{cos (𝑥)}]

\begin{matrix} 𝑓 (𝑥) = 𝑥^{2} 𝑔 (𝑥) = cos (𝑥) \\ 𝑓^{'} (𝑥) = 2 𝑥 𝑔^{'} (𝑥) = - sin (𝑥) \end{matrix}

\frac{𝑑}{𝑑 𝑥} [\frac{𝑥^{2}}{cos (𝑥)}] = \frac{2 𝑥 cos (𝑥) - 𝑥^{2} (- sin (𝑥))}{{[cos (𝑥)]}^{2}}

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [tan (𝑥)] & = \frac{𝑑}{𝑑 𝑥} [\frac{sin (𝑥)}{cos (𝑥)}] \\ = \frac{cos (𝑥) \cdot cos (𝑥) - sin (𝑥) \cdot - sin (𝑥)}{{cos}^{2} (𝑥)} \\ = \frac{{cos}^{2} (𝑥) + {sin}^{2} (𝑥)}{{cos}^{2} (𝑥)} \\ = \frac{1}{{cos}^{2} (𝑥)} \\ = {sec}^{2} (𝑥) \end{aligned}

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [cot (𝑥)] & = \frac{𝑑}{𝑑 𝑥} [\frac{cos (𝑥)}{sin (𝑥)}] \\ = \frac{cos (𝑥) \cdot cos (𝑥) - sin (𝑥) \cdot - sin (𝑥)}{{cos}^{2} (𝑥)} \\ = \frac{- {sin}^{2} (𝑥) - {cos}^{2} (𝑥)}{} \\ = - \frac{1}{{sin}^{2}} (𝑥) \\ = - {csc}^{2} (𝑥) \end{aligned}

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [sec (𝑥)] & = \frac{𝑑}{𝑑 𝑥} [\frac{1}{cos (𝑥)}] \\ = \frac{0 \cdot cos (𝑥) - 1 \cdot - sin (𝑥)}{{cos}^{2} (𝑥)} \\ = \frac{0 + 1 \cdot sin (𝑥)}{{cos}^{2} (𝑥)} \\ = \frac{sin (𝑥)}{{cos}^{2} (𝑥)} \\ = \frac{sin (𝑥)}{cos (𝑥)} \cdot \frac{1}{cos (𝑥)} \\ = tan (𝑥) \cdot sec (𝑥) \end{aligned}

\begin{aligned} \frac{𝑑}{𝑑 𝑥} [csc (𝑥)] & = \frac{𝑑}{𝑑 𝑥} [\frac{1}{sin (𝑥)}] \\ = \frac{0 \cdot sin (𝑥) - 1 \cdot cos (𝑥)}{{sin}^{2} (𝑥)} \\ = \frac{0 - 1 \cdot cos (𝑥)}{{sin}^{2} (𝑥)} \\ = \frac{- cos (𝑥)}{{sin}^{2} (𝑥)} \\ = - \frac{cos (𝑥)}{sin (𝑥)} \cdot \frac{1}{sin (𝑥)} \\ = cot (𝑥) \cdot csc (𝑥) \end{aligned}