#### Related to [[Limit]]
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- The *derivative* of any function is **another function that gives you the slope of the tangent** for any value of f(x).
- Although there are many rules we use to find derivatives the basic formula for finding a function's *derivative* from either the definition or graph is given below.
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## Derivative basics
##### *Conceptual definition*
- The derivative of a function describes the **slope of that function** at any given *x* value. Although derivatives are traditionally found using a limit there are a set of rules that allow one to easily convert any function into its derivative without the use of a limit.
$\text{The derivative of } f(x)\text{ is denoted as } f'(x) \text{ or }\frac{dy}{dx} $
>[!example]
>$\text{Visualizing the derivative}$
> ![[Tangent_function_animation.gif|350]]
##### *Mathematical definition*
- To find the derivative of a function without using any *shortcuts* or *derivative rules* one must use the equation from which those *rules* were originally created - the mathematical *definition of the derivative*. This equation is most commonly known as the **limit definition of the derivative.**
- The limit expression used to find derivatives is a slightly revised version of *the average rate of change formula*. Using the limit to evaluate the average velocity of a function across an infinitely small area gives us a function which effectively represents the **instantaneous rate of change at any given x value *also known as the derivative!***
$\text{Slope: }m=\frac{y_1-y}{x_1-x}\Longrightarrow m=\frac{f(x+h)-f(x)}{h}\space\space\text{\small Where }h=x_1-x$
>[!important]
>##### Derivative from definition
>- The derivative of a function gives its **instantaneous** velocity at any point, *x*. Therefore we must solve the equation above in a limit as *h (the distance between points averaged)* approaches 0.
>$\newline$
>$\text{\color{Bisque}\large Deffinition of derivative:}$
>$\newline$
> $\large\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=f'(x)=\frac{dy}{dx}$
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## Derivative rules
- The following are rules which one can follow to find the derivative of a function without using limits as outlined above.
##### Standard algebraic rules
>[!quote]
>| *Name* | *Rule* | *Example* |
>|-|-|-|
>| **Constant Rule** | $\text{}\space(C)'=0$ | $(3+2)'\rightarrow 0+0$
>| **Power rule** | $(x^n)' = nx^{n-1}$ | $(x^3 + 3x)'\rightarrow 3x^2+3$
>| **Multiple rule** | $[3f(x)]'=3f'(x)$ | $[3(x^2+2)]'\rightarrow 3(2x+0)$
>| **Sum/difference rule** | $[f(x)+g(x)]'=f'(x)+g'(x)$ | $(3x^2+2x)'\rightarrow (6x)+(2)$
>| **Product rule** | $[f(x)\cdot g(x)]'=f'(x)\cdot g(x)+f(x)\cdot g'(x)$ | $(2x\cdot 3x^2)'\rightarrow (2\cdot 3x^2+2x\cdot 6x)$
>| **Quotient rule** | $\left(\frac{f(x)}{g(x)}\right)'=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g(x)^2}$ | $\left(\frac{x^2}{3x}\right)'\rightarrow\frac{2x\cdot 3x-x^2\cdot 3}{(3x)^2}$
>| **Chain rule** | $f(g(x))'=f'(g(x))\cdot g(x)'$ | $\left( (2x)^{20}\right) '\rightarrow 20(2x)^{19}\cdot 2$
##### Trigonometric rules
>[!quote]
>
>Note that all trigonometric functions which contain a function inside must have the chain rule applied to them **in addition** to the rules outlined below. This is denoted as *x'*.
>
>| ***Name*** | ***Rule*** | ***Example*** |
>|-|-|-|
>| **Sin rule** | $sin(x)' = cos(x)\cdot x'$ | $sin(x^2)'\rightarrow cos(x^2)\cdot 2x$
>| **Cos rule** | $cos(x)' = -sin(x)\cdot x'$ | $cos(x^2)'\rightarrow -sin(x^2)\cdot 2x$
>| **Tan rule** | $tan(x)' = sec^2(x)\cdot x'$ | $tan(2x)'\rightarrow sec^2(2x)\cdot 2$
>| **Cot rule** | $cot(x)'=-csc^2(x)\cdot x'$ | $cot(2x)'\rightarrow -csc^2(2x)\cdot 2$
>| **Sec rule** | $sec(x)'=sec(x)tan(x)\cdot x'$ | $sec(x^3)'\rightarrow sec(x^3)tan(x^3)\cdot 3x^2$
>| **Csc rule** | $\csc(x)'=-csc(x)cot(x)\cdot x'$ | $csc(5x)'\rightarrow -csc(5x)cot(5x)\cdot 5$
##### Inverse trigonometric rules
>[!quote]
>| ***Name*** | ***Rule*** | ***Example***
>|-|-|-|
>| **Arcsine rule** | $sin^{\small -1}(x)'=\frac{x'}{\sqrt{1-x^2}}$ | $sin^{\small -1}(x^2)'\rightarrow \frac{2x}{\sqrt{1-x^4}}$
>| **Arccosine rule** | $cos^{\small -1}(x)'=\frac{-x'}{\sqrt{1-x^2}}$ | $cos^{\small -1}(3x)'\rightarrow \frac{-3}{\sqrt{1-9x^2}}$
>| **Arctan rule** | $tan^{\small -1}(x)'=\frac{x'}{1+x^2}$ | $tan^{\small -1}(x^2+3)\rightarrow \frac{2x}{1+(x^2+3)^2}$
>| **Arccsc rule** | $csc^{\small -1}(x)'=\frac{-x'}{\vert x\vert\sqrt{x^2-1}}$ | $csc^{\small -1}(-4x)'\rightarrow\frac{4}{4x\sqrt{16x^2-1}}$
>| **Arccot rule** | $cot^{\small -1}(x)'=\frac{-x'}{x^2+1}$ | $cot^{\small -1}(2x+1)'\rightarrow\frac{-2}{(2x+1)^2+1}$
>| **Arcsec rule** | $sec^{\small -1}(x)'=\frac{x'}{\vert x\vert\sqrt{x^2-1}}$ | $sec^{\small -1}(3x^2)'\rightarrow\frac{6x}{\vert 3x^2\vert\sqrt{9x^4-1}}$
##### Other rules
>[!quote]
>| ***Name*** | ***Rule*** | ***Example*** |
>|-|-|-|
>| **Natural log rule** | $ln(x)'=\frac{1}{x}\cdot x'$ | $ln(x^2+1)'\rightarrow\frac{1}{x^2+1}\cdot (2x)$
>| **Standard log rule** | $log_a(x)'=\frac{1}{x\cdot ln(a)}\cdot x'$ | $log_5(2x^2)'\rightarrow\frac{1}{2x^2\cdot ln(5)}\cdot (4x)$
>| **Euler's rule** | $(e^x)'=e^x\cdot x'$ | $\left( e^{2x}\right) '\rightarrow e^{2x}\cdot 2$
>| **Exponential function rule** | $\left( a^x\right)'=a^x\cdot (x'\cdot ln(a))$ $\textit{\small Use product rule to solve }(x'\cdot ln(a))$ | $\left( 4^{x^2}\right) '\rightarrow 4^{x^2}\cdot \left( 2x\cdot ln(4)+\frac{x^2}{4}\right)$
>
---
## Derivatives and graphs
##### Graph of the function $f(x)=\frac{dy}{dx}$
- If the function at point *x* is **increasing** *(has a positive slope)* that indicates that the derivative at that point is **positive**. Likewise if f(x) is **decreasing** *(has a negative slope)* the derivative is **negative**.
- If the function at point x is **flat** that indicates that its derivative at that point is 0.
- If a function's slope is constant *(linear)* its derivative in that interval is also constant *(linear)*.
##### Graph of the *derivative* $f'(x)$
- If the derivative is approaching 0 that indicates the function is **tapering off** since the function's tangent line is getting flatter. On the other hand, an increasing derivative *(in either the positive or negative direction)* indicates that the function's rate of change is **exponentially increasing**.
- Places where the derivative DNE *(a jump or hole in the graph)* are places where its base function DNE or forms an **"abrupt corner"** which causes the tangent's slope *(and therefore the derivative)* to instantly change.
- X values where the graph of the derivative forms a vertical asymptote are places where its base function **forms a vertical line** *(which causes the tangent's slope/derivative to approach infinity)*.
##### Graph of the second derivative $f''(x)$
- Intervals where a function's second derivative is positive are intervals where the function is **concave up**. Likewise, intervals where the second derivative is negative are intervals where **the function is concave down**.
- Places where a function's second derivative crosses the x axis are **inflection points**. Which are points where a function's **concavity changes**.
>[!example]
>##### Concave up vs down
>- Intervals where a function is concave up are intervals where its derivative is **increasing** *(includes decreasing negative derivatives)* or intervals where the **second derivative $f''(x)$ is positive**. Given a function's graph these are places where it is *U* shaped.
>- On the other hand intervals where a function is concave **down** are intervals where its derivative is **decreasing** or it's **second derivative $f''(x)$ is negative**. Visually these are places where a function looks like ∩.
>##### Inflection points
>- Inflection points are places where the **second derivative of a function crosses or touches the x axis** *("riding" the x axis does not count )*.
>- These are points where the concavity of the function changes.
>##### *Example:*
>![[Derivativepic3.png|850]]
>[!tip]
>##### Critical points *(KEY TERM)*
>- Critical points are places where the derivative of a function either **doesn't exist or is equal to zero** *(Note: intervals where $f'(x)$ equals 0 do not count as critical points)*. Critical points denote locations where the slope of a function changes abruptly and places where the slope's **sign changes** *(positive to negative or negative to positive)*.
>- Inflection points are places where a function's **second derivative** $f''(x)$ equals **zero**. They indicate when the graph of a function switches from being **concave down to concave up** or the other way around.
>##### Example: circles indicate *critical points*, squares indicate *inflection points:*
>![[Derivativepic2.png|850]]
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##### Minimums and maximums
- **LOCAL MAXIMA:** A function has **local minimums** at all values of x where the slope of it's tangent **switches from negative to positive**. In other words a function has local minimums at x values where it's derivative crosses the x axis *(from negative to positive)* or the base function stops decreasing and starts increasing.
</br>
- **LOCAL MINIMA:** Likewise, a function has a **local maxima** at any point where the slope of its tangent **changes sign from positive to negative**. These are also x values where the derivative crosses the x axis *(from positive to negative)* and the function stops increasing and begins to decrease.
</br>
- **GLOBAL MAX AND MIN:** The **global minimum** of a function is the **y** value of its **lowest critical point**. The **global maximum** of a function is the y value of its **highest critical point**.
</br>
>[!caution]
>##### Min max theorem
>- If you are investigating a function **that is bounded by some given interval** its important to recall that the **min and max values of the interval *count are critical points themselves*.**
>- Although these bounding values **cant count as a local min/max**, they can **count as the global min/max** if they represent the lowest and highest points in the **entire** interval.
></br>
>- As a result of this theorem we can conclude that a function **bounded by two points *(the interval)* will *always* have a global/local min and max.**
>---
> ![[Derivativepic5.png|350]]
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## Derivative applications
- Generally in physics functions show the relationship between **time and the position of an object**. By taking the derivative of *this position function* we create a new equation that relates **time and *velocity***. The **second derivative** of a *position function* relates **time and *acceleration***. *See more at [[Free fall motion]]*
##### *Mean value theorem*
- The mean value theorem is a theorem about derivatives and slope. **It states that the derivative of a function will equal the function's mean *(average)* slope on *any interval* exactly *one time*.** In other words, there is always a place on a function where the slope of a tangent is *equal* to the average slope on that interval.
- Mathematicians *used* this theorem to derive the fundamental theorem of calculus, which serves as the basis for [[Integration]]!
$\text{Mean value theorem: }f'(c)=\frac{f(b)-f(a)}{b-a}$
>[!quote]
>##### *Mean value theorem visual*
>![[Derivativemeanvaluetheorem.png|850]]
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## Second derivatives
- The second derivative of a function is the **derivative *of* the derivative function.** Like the *first derivative* the second derivative function denotes the slope of the first derivative function at any given point *x*. Calculating the **second derivative is exactly like calculating the first derivative, the same exact rules are applied**.
- Third, fourth fifth... derivatives are simply the **previous derivative function** that is - the third derivative is the derivative of the second derivative *(which is the derivative of the original function)*.
$\text{The second derivative of }f(x)\text{ is denoted as }f''(x) \text{ or }\frac{d^2y}{dx^2}$
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## Calculator
>[!cite]
>
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><iframe src="https://www.wolframalpha.com/calculators/derivative-calculator/"width="850" height="1200"></iframe>