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?

1 Answer
Feb 4, 2018

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.