Difference between revisions 102467107 and 102758250 on enwiki

:''For more background on this topic, see [[derivative]].''

===Example 1===
Consider ''f''(''x'') = 5:

: <math>f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{f(x+h)-5}{h} =  \lim_{h\rightarrow 0} \frac{(5-5)}{h} = \lim_{h\rightarrow 0} \frac{0}{h} = \lim_{h\rightarrow 0} 0 = 0</math>

The derivative of a [[constant function]] is [[0 (number)|zero]].

===Example 2===
Consider the graph of <math>f(x)=2x-3</math>. If the reader has an understanding of [[algebra]] and the [[Cartesian coordinate system]], the reader should be able to independently determine that this [[line (mathematics)|line]] has a slope of 2 at every point. Using the above quotient (along with an understanding of the [[limit (mathematics)|limit]], [[secant]], and [[tangent]]) one can determine the slope at (4,5):

:<math>
\begin{align}
f'(4) &= \lim_{h\to 0}\frac{f(4+h)-f(4)}{h}           \\
      &= \lim_{h\to 0}\frac{2(4+h)-3-(2\cdot 4-3)}{h} \\
      &= \lim_{h\to 0}\frac{8+2h-3-8+3}{h}            \\
      &= \lim_{h\to 0}\frac{2h}{h}                    \\
      &= 2
\end{align}
</math>

The derivative and slope are equivalent.

The derivative is difficult to comprehend for its douchebag properties. You see if you can fully comprehend a derivative, any at all, you are a douchebag.⏎
⏎
===Example 3===
Via differentiation, one can find the slope of a curve. Consider <math>f(x)=x^2</math>:

:{|
|-
|<math> f'(x)\, </math>
|<math>= \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} </math>
|-
(contracted; show full)\end{align}
</math>


[[Category:calculus]] [[Category:Mathematical notation]]

[[eo:Derivaĵo (ekzemploj)]]
[[fr:Exemples de calcul de dérivée]]