Differentiation from first principles query

Richard

I have a query surrounding differentiation from first principles.

I have to get around the problem that
since δx->0 and δy->0, therefore δy/δx ->0/0 which is meaningless.

I think that although δx->0 and δy->0, they never reach zero and hence one is not dividing by zero.

Is this correct, or am I missing something?

Michael Collins

Ah, very good question!

You are correct in that you are not dividing by zero. Since you are taking the limit, you want the value of the fraction as delta x -> 0, that is, as delta x goes to zero, not equal to zero remember. So if instead of putting in zero for delta x, you put in a small number say 0.001, and then see what the value is, then try a smaller number and see what the new value is, and keep trying this, you'll see that it "approaches" the limit - i.e. the derivitive in this case.

The informal defintion of a limit for functions of one variable is that if a function goes to the limit L, as x -> p, then this means that we can make the function as close as we want to L, by making x a close as is necessary to p.

I hope this kinda answers your question. This is something that's a bit tricky to grasp at first, and in my experience most people never even think of it. Also you say that 0/0 is meaningless, well sort of, it means you have an indeterminate state, the limit may or may not exist, you just can't tell from the form you've found.