# Derivatives

###### tags: `math`

## Algebraic Functions

**Note:** $a$, $c$, and $n$ are constants; $u$ and $v$ are functions of $x$.

$$
\begin{aligned}
\frac{\text{d}}{\text{d}x}(c) &= 0 \\
\frac{\text{d}}{\text{d}x}(x) &= 1 \\
\frac{\text{d}}{\text{d}x}(x^n) &= nx^{n-1} \\
\\
\frac{\text{d}}{\text{d}x}(au) &= au'\\
\frac{\text{d}}{\text{d}x}(u \pm v) &= u' \pm v' \\
\frac{\text{d}}{\text{d}x}(u \cdot v) &= u'v+uv' \\
\frac{\text{d}}{\text{d}x}\left(\frac{u}{v}\right) &= \frac{u'v-uv'}{v^2} \\
\end{aligned}
$$

## Trigonometric Functions

$$
\begin{aligned}
\frac{\text{d}}{\text{d}x}(\sin x) &= \cos x \\
\frac{\text{d}}{\text{d}x}(\cos x) &= -\sin x \\
\frac{\text{d}}{\text{d}x}(\tan x) &= \sec^2 x \\
\\
\frac{\text{d}}{\text{d}x}(\csc x) &= -\csc x \cot x\\
\frac{\text{d}}{\text{d}x}(\sec x) &= \sec x \tan x\\
\frac{\text{d}}{\text{d}x}(\cot x) &= -\csc^2 x\\
\end{aligned}
$$

## Derivative Applications

### Tangent Lines (Garis Singgung)

#### Slope of tangent line (gradien garis singgung)

The slope of a line tangent to $f(x)$ at $x = a$ is
$$
m = f'(a).
$$

#### Relation between two lines

Let there be 2 lines with slopes $m_1$ and $m_2$. The relation between these two lines is
$$
\begin{aligned}
\mathbf{Parallel} \text{ (sejajar): }& m_1 = m_2 \\
\mathbf{Perpendicular} \text{ (tegak lurus): }& m_1 \cdot m_2 = 1 \text{ or } m_2 = -\frac{1}{m_1}
\end{aligned}
$$

#### Equation of a line

- If you know the gradient of a line, say $m$, and the line passes through point $(0, c)$ on the $y$-axis:
$$
y = mx + c
$$

- If you know the gradient of a line, say $m$, and a point the line passes through, say $(x_1, y_1)$:
$$
y - y_1 = m (x - x_1)
$$

- If you know two points the line passes through, say $(x_1, y_1)$ and $(x_2, y_2)$:
$$
\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}
$$

### Increasing and Decreasing, Stationary Points

Let $f(x)$ be a function of $x$.

- The function is **increasing** (naik) in the interval where
$$
f'(x) > 0
$$

- The function is **decreasing** (turun) in the interval where
$$
f'(x) < 0
$$

- The function has a **stationary point** (titik stasioner) in the interval where
$$
f'(x) = 0
$$

- The function is **non-increasing** (tidak pernah naik) in the interval where
$$
f'(x) \leq 0
$$

- The function is **non-decreasing** (tidak pernah turun) in the interval where
$$
f'(x) \geq 0
$$