As far as I can see, you introduce a new denominator but you don't à priori know if that expression isn't zero. In this case, if g(x) = g(c), you have a problem. If g(x)-g(c) is not zero, you can indeed take the limit and you're done. However, to allow the case where g(x) = g(c), it's a bit more subtle. This problem can be resolved by defining another function, which doesn't change the case where g(x) doesn't equal g(c), but which takes the desired derivative by definition in the case of g(x) = g(c).