35 lines
580 B
Markdown
35 lines
580 B
Markdown
|
# Proof of x² = 2x
|
||
|
|
|
||
|
|
|
||
|
|
$$
|
||
|
|
\begin{flalign}
|
||
|
|
& \frac{d}{dx}(x^2) = 2 &\\\
|
||
|
|
\\
|
||
|
|
& f(x) = x^2 &\\\
|
||
|
|
\\
|
||
|
|
&f'(x) = \lim_{x \to 0} \frac{f(x+h) - f(x)}{h} \\
|
||
|
|
\\
|
||
|
|
&\text{So what is }f(x+h)?\\
|
||
|
|
&\text{We just replace the x in the base formula with }(x+h)\\
|
||
|
|
&f(x+h) = (x+h)^2\\
|
||
|
|
\\
|
||
|
|
&f'(x) = \lim_{x \to 0} \frac{(x+h)^2-x^2}{h} \\
|
||
|
|
&f'(x) = \lim_{x \to 0} \frac{x^2+2xh+h^2-x^2}{h} \\
|
||
|
|
&f'(x) = \lim_{x \to 0} \frac{2xh+h^2}{h} \\
|
||
|
|
&f'(x) = \lim_{x \to 0} \frac{h(2x+h)}{h} \\
|
||
|
|
&f'(x) = \lim_{x \to 0} 2x+h \\
|
||
|
|
|
||
|
|
\end{flalign}
|
||
|
|
$$
|
||
|
|
|
||
|
|
|
||
|
|
```desmos-graph
|
||
|
|
left=-2; right=2;
|
||
|
|
bottom=-2; top=2;
|
||
|
|
---
|
||
|
|
y=x^2
|
||
|
|
y=2x
|
||
|
|
```
|
||
|
|
|
||
|
|
|