gjm has read the note I linked; I suggest you do the same. That is what a link is for.

I wish I hadn't made my comment about precision, which was too nitpicking and unhelpful. But as long as we're being snippy with each other:

To be excruciatingly precise: You just said you were being precise, then said "Let x be a differentiable real function." That isn't precise; you need to specify right there that it's a function of t. If you'd said the link stated it precisely, that would be different.

I admit that I would have interpreted it correctly by making the most-favorable, most-reasonable interpretation and assuming x was a function of t. But, because of the sorts of things I usually see done with x and t, I assumed that x was a function of time, and the function of interest was some function of x(t), and I jumped to the conclusion that you meant to say "Let f(x) be a differentiable real function." Which I would not have done had you in fact been precise, and said "Let x(t) be a differentiable real function."