Kenneth S. answered • 09/22/17
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You must mean that f(x) = (3x-6)1/3 + 5 and so its derivative will be (3)(1/3)(3x-6)-2/3
which is putting the cube root of 3x-6 into a denominator (and then squaring it).
So you cannot permit 3x-6 being zero; therefore f is not differentiable at x=2.
Kenneth S.
Actually, I applied the POWER rule but I forgot to also apply the Chain rule, so that the 1/3 gets canceled by the 3 factor coming from differentiation of that which was being raised to the original exponent 1/3. Nevertheless, the key observation, that 3x-6 be prevented from becoming zero, is correct. Sorry about that careless omission.
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