Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision		Previous revision
math:derivative [2021/09/15 19:29] – created epixmath:derivative [2021/09/16 15:01] (current) – made image look better epix
Line 2: Line 2:
 The derivative of a function (say $f(x)$) is a function of the slope of the tangent line at any given point. It is a generic slope formula for any x value, or instantaneous rate of change. The output of the derivative can be used to make tangent lines. Its notation is $f'(x)$ or $\frac{d}{dx} f(x)$. The derivative of a function (say $f(x)$) is a function of the slope of the tangent line at any given point. It is a generic slope formula for any x value, or instantaneous rate of change. The output of the derivative can be used to make tangent lines. Its notation is $f'(x)$ or $\frac{d}{dx} f(x)$.
  
 +===== When is a Derivative Valid? =====
 +A derivative is valid when the function is [[math:limits|continuous]] and when it is differentiable, meaning that both sides of a function approaches the same slope at any given point. If not, this means that there is either a cusp, corner, or vertical tangent present.
 +
 +{{:math:der6.gif|}}
 +
 +  * A cusp is present when each side of a function approaches a vertical slope, which Does Not Exist.
 +  * A corner is present when there is a sharp change in slope, which prevents a continuous derivative.
 +  * A vertical tangent means that each side of the slope function approaches infinity from a different direction.
 +  * A discontinuity means that the slope function is also has a discontinuity, which prevents a continuous derivative.
 ===== Calculating the Derivative ===== ===== Calculating the Derivative =====
 +[{{ :math:tangent_animation.gif?400|$\Delta x$ in this visualization is the same as $h$ in the definition of a limit}}]
 Using the slope formula $m = \frac{y_2-y_1}{x_2-x_1}$, you can derive the definition of a derivative using a limit. Using the slope formula $m = \frac{y_2-y_1}{x_2-x_1}$, you can derive the definition of a derivative using a limit.
  
Line 10: Line 20:
  
 ==== Using the Alternative Form ==== ==== Using the Alternative Form ====
-FIXME alternative form! and add a graphic for the definition of a derivative.+Using the definition of a derivative is a pain in the ass. There is an alternative definition using the same slope formula for the slope at one specific point that you can use instead if you don't need the derivative function. 
 + 
 +$$ \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a) $$ 
 + 
 +  * $a$ is the point where you wish to find the tangent line 
 + 
 +This is much simpler to compute, however it __only__ works for the one value $a$. Don't mix it up and think you can plug in any value once you simplify to get the slope at any point! 
 + 
 +==== This Sucks I Want Out! ==== 
 +No worries! Calculating limits is hard, that's why there are [[math:derivative_rules|rules]] to save yourself time when calculating derivatives. Hopefully you'll never have to use the definition of a derivative to find it.

Trace:

Navigation

bruh

QR Code
QR Code math:derivative (generated for current page)