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ROAM_ALIASES	Differentiable Differentiation Derivative

Differentiation

In calculus, differentiation is the process of defining the "rate of change at a point" or the "instantaneous rate of change" on a curve as the slope of the limit of the secant lines between increasingly close points.

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https://www.desmos.com/calculator/btelpebrlh
A function $f$ is differentiable at $c$ if it is smooth and does not contain any break, angle, or cusp.[cite:@enwiki:1294653445]
The equation for the tangent line at $a$ is $y = (f^{'} a) (x - a) + f a$ .
Beautifully, the derivative can be seen as a higher-order function, and is an example of a limit returning a non-numeric object.
- $deriv : (ℝ \to ℝ) \to ℝ \to ℝ$ .
- TODO: is a derivative an endofunctor on some category??? see https://ncatlab.org/nlab/show/chain+rule

uhh

for any function $f$ that is differentiable at $x$ :

$\lim_{h \to 0} \frac{f (x + h) - f x}{h} = \lim_{h \to x} \frac{f h - f x}{h - x}$

Definition

A natural definition for the derivative arises from the intuition of the slopes of secant lines through two increasingly-close points by taking the limit:

$\frac{d}{d x} f x = \lim_{h \to 0} \frac{f (a + h) - f a}{h} .$

Basic derivatives

Recall pointwise function arithmetic notation for this section!

Derivatives of trigonometric functions

Calculus classes will often expect you to keep the derivatives of trigonometric functions in working memory.[cite:@enwiki:1303550916]

These equations only hold when using radians.

\begin{tabular}{p{3.4cm}p{3.4cm}} \begin{gather*} \sin' = \cos \end{gather*} & \begin{gather*} \cos' x = - \sin x \end{gather*} \\ \begin{gather*} \tan' = \sec^2 \end{gather*} & \begin{gather*} \cot' x = -\csc^2 x \end{gather*} \\ \begin{gather*} \sec' = \sec \times \tan \end{gather*} & \begin{gather*} \csc' x = - (\csc x)(\cot x) \end{gather*} \\ \begin{gather*} \arcsin' x = \frac{1}{\sqrt{1-x^2}} \end{gather*} & \begin{gather*} \arccos' x = -\frac{1}{\sqrt{1-x^2}} \end{gather*} \\ \begin{gather*} \arctan' x = \frac{1}{x^2 + 1} \end{gather*} & \begin{gather*} \arccot' x = -\frac{1}{x^2 + 1} \end{gather*} \\ \begin{gather*} \arcsec' x = \frac{1}{\lvert x \rvert \sqrt{x^2 - 1}} \end{gather*} & \begin{gather*} \arccsc x = -\frac{1}{\lvert x \rvert \sqrt{x^2 - 1}} \end{gather*} \end{tabular} ParseError: No such environment: tabular at position 7: \begin{̲t̲a̲b̲u̲l̲a̲r̲}̲{p{3.4cm}p{3.4c…

Repeated differentiation of sine and cosine is cyclic: $\begin{matrix} \frac{d}{d x} \sin x & = \cos x \\ \frac{d}{d x} \cos x & = - \sin x \\ \frac{d}{d x} (- \sin x) & = - \cos x \\ \frac{d}{d x} (- \cos x) & = \sin x \\ ⋮ \end{matrix}$ This property massively simplifies higher derivatives of the functions: $\frac{d^{n + 4}}{d x^{n + 4}} \sin x = \frac{d^{n}}{d x^{n}} \sin x .$

References

[cite:@enwiki:1309366243]