Differentiability definition

Question

I am trying to understand the definition of differentiation, I am working on some problems and ran across this one:

If f+g is differentiable at x = a and f is differentiable at x = a, then g is differentiable at x = a

I think this is true because the addition operation shouldn't change it, but I find it difficult to use the definition to actually prove this. Like it seems like this should just work by properties but I am not sure.

Answer 1

Note that $$g=(f+g)-f$$

Thus it is easier to prove that the difference of two differentiable functions is differentiable first and apply the results to your functions.

Good news is that the proof for difference is almost identical to the proof for sum.