For what values of x is the function #f(x)=abs(x^2-9)# differentiable?

Although #x^2-9# is both continuous and differentiable everywhere the same is not true for #|x^2-9|#, which is continuous everywhere but not differentiable at the transition between positive and negative.

This transition occurs when:

# |x^2-9| = 0 => x^2-9=0 #
# " "=> x^2=9#
# " "=> x = +- 3#

Hence the function #f(x)=|x^2-9|# is differentiable everywhere with the exception of #x=+-3#

graph{|x^2-9| [-20.04, 19.96, -3.16, 16.84]}