From 2711f7e1cd57386daa47af3907d9b5f124bdebcb Mon Sep 17 00:00:00 2001
From: arc <zleyyij@users.noreply.github.com>
Date: Thu, 30 Jan 2025 09:18:43 -0700
Subject: [PATCH] vault backup: 2025-01-30 09:18:43

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+A derivative can be used to describe the rate of change at a single point, or the *instantaneous velocity*.
+
+The formula used to calculate the average rate of change looks like this:
+$$ \dfrac{f(b) - f(a)}{b - a} $$
+Interpreting it, this can be described as the change in $y$ over the change in $x$.
+
+- Speed is always positive
+- Velocity is directional
+
+As the distance between the two points $a$ and $b$ grow smaller, we get closer and closer to the instantaneous velocity of a point. Limits are suited to describing the behavior of a function as it approaches a point.
+
+If we have the coordinate pair $(a, f(a))$, and the value $h$ is the distance between $a$ and another $x$ value, the coordinates of that point can be described as ($(a + h, f(a + h))$. With this info:
+- The slope of the secant line can be described as $\dfrac{f(a + h) - f(a)}{a + h - a}$, which simplifies to $\dfrac{f(a + h) - f(a)}{h}$. 
+- The slope of the *tangent line* or the *instantaneous velocity* can be found by taking the limit of the above function as the distance ($h$) approaches zero:
+$$\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h}$$
+The above formula can be used to find the *derivative*. This may also be referred to as the *instantaneous velocity*, or the *instantaneous rate of change*.
+# Line Types
+## Secant Line
+A **Secant Line** connects two points on a graph.
+
+A **Tangent Line** represents the rate of change or slope at a single point on the graph.
+
+# Notation
+Given the equation $y = f(x)$, the following are all notations used to represent the derivative of $f$ at $x$:
+- $f'(x)$
+- $\dfrac{d}{dx}f(x)$
+- $y'$
+- $\dfrac{dy}{dx}$
+- $\dfrac{df}{dx}$
+- "Derivative of $f$ with respect to $x$"
+
+# Functions that are not differentiable at a given point
+- Where a function is not defined
+- Where a sharp turn takes place
+- If the slope of the tangent line is vertical
+
+# Higher Order Differentials
+- Take the differential of a differential
+
+Using the definition of a derivative to determine the derivative of $f(x) = x^n$, where $n$ is any natural number.
+
+$$ f'(x) = \lim_{h \to 0} \dfrac{(x + h)^n - x^n}{h} $$
+- Using pascal's triangle, we can approximate $(x + h)^n$
+```
+    1
+   1 1
+  1 2 1
+ 1 3 3 1
+1 4 6 4 1 
+```
+
+- Where $n = 0$: $(x + h)^0  = 1$
+- Where $n = 1$: $(x +h)^1 = 1x + 1h$
+- Where $n = 2$: $(x +h)^2 = x^2 + 2xh + h^2$
+- Where $n = 3$: $(x + h)^3 = 1x^3h^0 + 3x^2h^1 + 3x^1h^2 + 1x^0h^3 = 1x^3 + 3x^2h + 3xh^2 + 1h^3$
+
+Note that the coefficient follows the associated level of Pascal's Triangle (`1 3 3 1`), and $x$'s power decrements, while $h$'s power increments. The coefficients of each pair will always add up to $n$. Eg, $3 + 0$, $2 + 1$, $1 + 2$, and so on. The **second** term in the polynomial created will have a coefficient of $n$.