To William: on dy/dx and FTC

You have asked good questions.

Given x = x(t), y=y(t) and x=f(y). i.e.,

x(t) = f(y) where y = y(t).

By Chain Rule,

x'(t) = f'(y)\cdot y'(t).


x'(t), y'(t) – derivative of function x(t), y(t) with respect to t respectively.

f'(y) – derivative of function f(y) with respective to y.

Suppose y'(t) \ne 0, then

\frac{x'(t)}{y'(t)} = f'(y).

Written in Leibniz’s notation, \frac{\frac{dx}{dt}}{\frac{dy}{dt}} = \frac{df}{dy}.

Even though Leibniz’s notation is convenient at times, it often gives the WRONG idea that the derivative of a function is the ratio of two mysterious quantities (dy, dx) that are not zero and yet, smaller than any positive numbers.

So saying “split the derivative” is correct (from looking at the Chain Rule) since derivative implies a process of taking the limit:

f'(t) = \lim\limits_{\delta t \to  0} \frac{\delta f}{\delta t}= \lim\limits_{\delta t \to 0}\frac{f(t +\delta t) - f(t)}{\delta t},

not simply \frac{\delta f}{\delta t}.

To see how antiderivative and definite integral are related. one must dive into the proof of the Fundamental Theorem of Calculus (For example, see “A Sprint to FTC“). After that, you can decide for yourself how the concepts of rate of change and “area under the curve” are logically linked.