When finding out the domain of differentiability of a function do you have to use $\lim \limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$?

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Usually I check first if it's continuous at the given point, then take the derivative (if it has several branches I take the derivative of each branch) and then put the value there and see if the limit on the left and the right of the point match. Is this correct? Or do I have to use this formula specifically? Isn't it equivalent?