Linearity of Differential Operators: Is it valid to say that $\frac{d}{dy} |g'(y)| = |g''(y)|$?

Question

Since the differential operator is linear, is it valid to say that $\dfrac{d}{dy} |g'(y)| = |g''(y)|$?

I would appreciate it if someone would please clarify this.

Answer 1

Example. $g(x) = \sin x$. Then $g'(x) = \cos x$, $g''(x) = -\sin x$. Here are graphs,

$\displaystyle\frac{d}{dx}|g'(x)|$

$|g''(x)|$

Are they the same?