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education/math/MATH1210 (calc 1)/Derivatives.md

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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 = 1