RUMUS DASAR
| F(x) | F'(x) |
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(- \sin x\) |
| \(\tan x\) | \(\sec^2 x\) |
| \(\cot x\) | \(- \csc^2 x\) |
| \(\sec x\) | \(\sec x \tan x\) |
| \(\csc x\) | \(- \csc x \cot x\) |
ATURAN BERANTAI
\(\dfrac {dy}{dx} = \dfrac {dy}{du} \:.\: \dfrac {du}{dx}\)
Contoh 01
Tentukan turunan pertama dari \(y = \sin \: (2x + 5)\)
Cara pemisalan
\begin{equation*} \begin{split} & u =2x + 5 \\\\ & \frac {du}{dx} = 2 \end{split} \end{equation*}
\begin{equation*} \begin{split} & y = \sin u \\\\ & \frac {dy}{dx} = \frac {dy}{du} \:.\: \frac {du}{dx} \\\\ & \frac {dy}{dx} = \cos u \:.\: 2 \\\\ & \frac {dy}{dx} = 2 \:.\: \cos u \\\\ & \frac {dy}{dx} = \bbox[5px, border: 2px solid magenta] {2 \:.\: \cos \: (2x + 5)} \end{split} \end{equation*}
Cara langsung
\begin{equation*} \begin{split} & y = \sin \: ({\color {red} 2x + 5}) \\\\ & {\color {blue} \text{turunan dari } 2x + 5 \text{ adalah } 2} \\\\ & {\color {blue} \text{turunan dari } \sin \: (2x + 5) \text{ adalah } \cos \: (2x + 5)} \\\\ & \frac {dy}{dx} = \bbox[5px, border: 2px solid magenta] {2 \:.\: \cos \: (2x + 5)} \end{split} \end{equation*}
Contoh 02
Tentukan turunan pertama dari \(y = \cos^3 x\)
Cara pemisalan
\begin{equation*} \begin{split} & u =\cos x \\\\ & \frac {du}{dx} = - \sin x \end{split} \end{equation*}
\begin{equation*} \begin{split} & y = u^3 \\\\ & \frac {dy}{dx} = \frac {dy}{du} \:.\: \frac {du}{dx} \\\\ & \frac {dy}{dx} = 3 \:.\: u^2 \:.\: -\sin x \\\\ & \frac {dy}{dx} = -3 \:.\: \sin x \:.\: u^2 \\\\ & \frac {dy}{dx} = \bbox[5px, border: 2px solid magenta] {-3 \:.\: \sin x \:.\: \cos^2 x} \end{split} \end{equation*}
Cara langsung
\begin{equation*} \begin{split} & y = \cos^3 x \\\\ & {\color {blue} \text{turunan dari } \cos x \text{ adalah } - \sin x} \\\\ & \frac {dy}{dx} = - \sin x \:.\: 3 \:.\: \cos^2 x \\\\ & \frac {dy}{dx} = \bbox[5px, border: 2px solid magenta] {-3 \:.\: \sin x \:.\: \cos^2 x} \end{split} \end{equation*}