notes/Resources/mathematics/derivation/lim-proof.md

# Proof of x² = 2x


```latex
\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
```