Are there any functions which are unable to be differentiated?

Question

As in, say I had a function #f(x)#, and the derivative #f'(x)#. Is there any function for #f(x)# where the equation for #f'(x)# cannot be worked out using any differentiation rules and instead have to be done graphically.

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Answer 1 · 2017-12-16T16:39:40

Perhaps this is the kind of thing you're wondering about.

#f(x)={(x^2sin(1/x), "if", x != 0) ,(0,"if",x = 0):}#.

#f# can be differentiated by the chain rule at all #x != 0#, but to show that #f'(0)=0# we use (need?) the definition of derivative.

Graphical techniques are only as accurate as our ability to graph and read the graph.