Graph and derivative of y=x I'm having trouble understanding why the graph of $y=x$ is different from the graph of $y = \sqrt{ x^2 }$. Aren't both equations the same once you simplify the second one? And isn't the derivative of each the same?
It's different. $$y=\sqrt{x^2}=|x|$$ that is different from $y=x$
The derivate of $y=x$ is $y=1$ while the derivate of $y=|x|$ is $y=1 $ for $x&gt;0$ and $y=-1$ for $x&lt;0$
Consider $x=-1$:

$
\begin{align}
x&amp;=-1 \\
\sqrt{x^2}&amp;=\sqrt{(-1)^2} =\sqrt{1}=\pm1
\end{align}
$



Every root has two solutions: $\sqrt{a}=\pm a$, for example
$2 \cdot 2 = -2 \cdot -2 = 4$

Because a function can only have one $y$ for every $x$, one solution (the negative) has to be omitted. This causes the graph of $y=\sqrt{x^2}$ to always be positive.

If you would plot a graph with both the positive and negative solution of $y=\sqrt{x^2}$, it would look like $y=x$ and $y=-x$ united.