Sorry if I was sounding harsh - I appreciate that you are an A level student and this must be frustrating. If you'd said something like "is this OK as a plausible argument why the derivative of sin x is cos x?" then I would probably not have commented. My real concern is whether something like this is being presented as a 'proof' in a textbook, because it's really not precise enough to be called a proof
The problem is really with the innocent-looking statement "
sinh≈h when h is small" (I'm using h instead of delta x just to simplify my typing!). We all tend to look at this and say "Oh yes, sin h is approximately equal to h", but what does that actually mean mathematically? For example, it's true that
x2≈x when x is close to 0 because both quantities are close to 0, but that doesn't mean we can deduce that
x2/x→1 as x->0 - we know that the limit is in fact 0.
What we need here is actually that (sin h)/h -> 1 as h->0 and this is in fact the case, provided that h is measured in radians.
For info, here's what I regard as the 'standard' A level proof that the OP needs - taken from the 'bible' Bostock & Chandler. I assumed it was included in all standard A level texts,
but perhaps I'm just terribly naive