Question

What does derivative of y with respect to x mean?

if im deriving an equation for `dy/dx` , can the final answer have other variables (such as u), or does it have to be the derivative of y with only x variables?

Answer 1

See below.

Explanation:

You have really asked two different questions.

The first one: "What does derivative of y with respect to x mean?"

If we have some function `y =f(x)` that is diffenentiable. Then

`dy/dx = lim_(deltax->0) (f(x+deltax)-f(x))/(deltax)`

At it's simplest, `dy/dx` measures the rate of change or instantaneous slope of `y=f(x)` at the point `x`. [Thanks due to @Steve M in comment below]

The second one:

This question depends on the nature of `u`

(i) If `u` is some function of `x` then it must be "undone" when expressing `dy/dx`. E.g. the chain rule states that if `y=f(u(x))`

`dy/dx = dy/(du) * (du)/dx = f'[u(x)] * u'(x)`

(ii) If `u` is indpendent of `x` and not a variable in its own right, such as a constant, it can stand as it is.

(iii) If `u` is a another variable, independent of `x` (E.g. `y = f(x,u)`) we are in realm partial dfferentiation where `dy/dx` would not be applicable.

Hope this helps.