Derivative

Sidedness

Left-side derivative at $a$ :

f_{-}^{'} (a) = lim_{h \to 0^{-}} \frac{f (a + h) - f (a)}{h}

Right-side derivative at $a$ :

f_{+}^{'} (a) = lim_{h \to 0^{+}} \frac{f (a + h) - f (a)}{h}

$f'(x) = L \iff f'_-(x) = f'_+(x) = L$

Implicit differentiation

Find $y^{'}$ / $x^{'}$ by:

Considering $y$ / $x$ as a function of $x$ / $y$
Differentiating as normal using the Chain Rule

$y^{'}$ and $x^{'}$ may depend on two variables.
To evaluate $y^{'}$ at $(x_{0}, y_{0})$ , we write

y^{'} (x_{0}, y_{0}) = \dots

[!example]+ Find $\frac{d y}{d x}$ of $x^{2} + x y + y^{2} = 1$ .
$\begin{aligned} x^{2} + x y + y^{2} & = 1 \\ \frac{d}{d x} (x^{2} + x y + y^{2}) & = \frac{d}{d x} 1 \\ 2 x + y + x y^{'} + 2 y y^{'} & = 0 \\ y^{'} (x + 2 y) & = - (2 x + y) \\ y^{'} & = - \frac{2 x + y}{x + 2 y} \end{aligned}$

[!example]- Find $\frac{d x}{d y}$ of $x^{2} + x y + y^{2} = 1$ .
$\begin{aligned} x^{2} + x y + y^{2} & = 1 \\ \frac{d}{d x} (x^{2} + x y + y^{2}) & = \frac{d}{d x} 1 \\ 2 x x^{'} + x^{'} y + x + 2 y & = 0 \\ x^{'} (2 x + y) & = - (x + 2 y) \\ x^{'} & = - \frac{x + 2 y}{2 x + y} \end{aligned}$

Find the slope of the tangent line to the curve $x y \ln (2 - y x) = 0$ .

\begin{aligned} x y \ln (2 - x y) & = 0 \\ \frac{d}{d x} (x y \ln (2 - x y)) & = \frac{d}{d x} 0 \\ y \ln (2 - x y) + x y^{'} \ln (2 - x y) + x y \cdot \frac{(2 - x y)^{'}}{2 - x y} & = 0 \\ y \ln (2 - x y) + x y^{'} \ln (2 - x y) - x y \cdot \frac{y + x y^{'}}{2 - x y} & = 0 \end{aligned}

Plug $(1, 1)$ :

\begin{aligned} 1 \ln (2 - 1 \cdot 1) + 1 y^{'} (1, 1) \ln (2 - 1 \cdot 1) - 1 \cdot 1 \frac{1 + 1 y^{'} (1, 1)}{2 - 1 \cdot 1} & = 0 \\ - (1 + y^{'} (1, 1)) & = 0 \\ y^{'} (1, 1) & = - 1 \end{aligned}

Properties & theorems

$\frac{d}{d x} c = 0$
$\frac{d}{d x} (c \cdot f (x)) = c \cdot f^{'} (x)$
$\frac{d}{d x} (f (x) \pm g (x)) = f^{'} (x) \pm g^{'} (x)$
Product Rule: $\frac{d}{d x} (f (x) \cdot g (x)) = f (x) g^{'} (x) + g (x) f^{'} (x)$
Quotient Rule: $\frac{d}{d x} \frac{f (x)}{g (x)} = \frac{g (x) f^{'} (x) - f (x) g^{'} (x)}{(g (x))^{2}}, provided g (x) \neq 0$
Power Rule: $\frac{d}{d x} x^{n} = n x^{n - 1}$
$\frac{d}{d x} \sin x = \cos x$
$\frac{d}{d x} \cos x = - \sin x$
$\frac{d}{d x} \tan x = \sec^{2} x$
$\frac{d}{d x} \sec x = \tan x \sec x$
$\frac{d}{d x} \csc x = - \cot x \csc x$
$\frac{d}{d x} \cot x = - \csc^{2} x$

Contributors

Ajitani Hifumi

Changelog

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Derivative ​

Sidedness ​

Implicit differentiation ​

Properties & theorems ​