# The derivative

Hi,
I'm just wondering when finding the derivative when do you use dy/dx and when do you use the
f'(x)= lim h-->0 f(x+h)-f(x) /h
Original post by Animal lover :))
Hi,
I'm just wondering when finding the derivative when do you use dy/dx and when do you use the
f'(x)= lim h-->0 f(x+h)-f(x) /h

f'(x) and dy/dx are just different notation of the same thing. dy/dx was down to Liebniz and f'(x) was Lagrange. You could go into how f' was developed along with functions and dy/dx was when you thought of y as a variable (which depends on x). Newton used dot (fluxions) and Euler used D.

Edit - as per TypicalNerds post, if youre asking when do you prove a derivative from first principles (the f'(x) part of your question) of when you use standard results to evaulate the derivative, then its pretty much that. Youd use the f'(x) = ... definition when youre asked to show something from first principles. Otherwise youd use the standard stuff (derivative of exponential, power, chain rule, ...).
(edited 5 months ago)
Original post by Animal lover :))
Hi,
I'm just wondering when finding the derivative when do you use dy/dx and when do you use the
f'(x)= lim h-->0 f(x+h)-f(x) /h

The formula here is used when the question asks you to prove a derivative from first principles. Though mqb2766 is absolutely right that dy/dx and f’(x) are just alternative ways of representing the derivative of a function
Original post by TypicalNerd
The formula here is used when the question asks you to prove a derivative from first principles. Though mqb2766 is absolutely right that dy/dx and f’(x) are just alternative ways of representing the derivative of a function

You will also often need to use "first principles" if the function has a "split definition". E.g. if $f(x) = x^2 \sin(1/x)\, x\neq 0, f(0)=0$ then finding f'(0) basically requires a "first principles" approach.
(edited 5 months ago)
Original post by DFranklin
You will also often need to use "first principles" if the function has a "split definition". E.g. if $f(x) = x^2 \sin(1/x)\, x\neq 0, f(0)=0$ then finding f'(0) basically requires a "first principles" approach.

Perhaps, though I was working under the assumption that the OP is doing ordinary A level maths and so they most probably wouldn’t need to consider such a case (though my memory could be terrible and such instances may be covered on the course)