If $f(x)$ is differentiable, then isn't $|f(x)|$ not differentiable where $f(x)=0$ and differentiable everywhere else?

Question

Problem 2-13 (d) of Spivak's Calculus on Manifolds is:

But isn't $|f(t)|$​ always differentiable everywhere that $f(t)$ is differentiable and not equal to $0$? So isn't it impossible to find a differentiable function $f(t)$ such that $|f(t)|$ is nowhere differentiable?