I observe that for $x = 0$, $f'(x)$ is $0$ both from the first
and from the second form of $f(x)$, that is on both sides of the
origin. So the first derivative $f'(0)=0$ exists at $x=0$. Hence
the graph of $f(x)$ has the tangent at $y=0$ at the origin.
Now, I calculate the second derivative:
$$\eqalign{ f''(x) &= 0\ for\ x < 0 \cr &= 4\ for \
x> 0.}$$
Hence the second derivative does not exist at the origin
because on the left the limiting value of $f''(x)$ as $x\to 0$ is
$0$ whereas on the right the limiting value of $f''(x)$ as $x\to 0$
is $4$. So there isn't a unique tangent to the graph of $f'(x)$ at
$x = 0$.