Answer
Since the derivatives are not equal from the left and the right, the $f$ is not differentiable at $1$.
We can see a sketch of $f$ and $f'$ below.
Work Step by Step
$f(x) = x^2+1~~~$ if $x \lt 1$
$f(x) = x+1~~~$ if $x \geq 1$
As $x \to 1^-$, then $~~f'(x) = 2x~~$ and the derivative approaches the value of 2
As $x \to 1^+$, then $~~f'(x) = 1~~$
Since the derivatives are not equal from the left and the right, $f$ is not differentiable at $1$
We can see a sketch of $f$ and $f'$ below.