Differentiation Notations
================================


Leibniz's Notation
-------------------

The derivative of a function :math:`f` at :math:`x` is given by :math:`\lim\limits_{h\to0} \frac{f(x+h)-f(x)}{h}`.
The Leibniz's notation expresses the derivative of :math:`f` as :math:`\frac{dy}{dx}` with :math:`y=f(x)`.
More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__.

.. note::
   The following :math:`\frac{dx}{dt}` means :math:`x` is a function of :math:`t` such as :math:`x=f(t)`.
   See explanations `here <https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-4/v/differentiating-related-functions-intro>`__.

:math:`\frac{d}{dx}` is an operator not a quotient!
Although it behaves like a quotient. In fact it is a limit:

.. math::

   \lim\limits_{h\to0} \frac{f(x+h)-f(x)}{h} \ne \frac{\lim\limits_{h\to0} \left(f(x+h)-f(x)\right)}{\lim\limits_{h\to0} h}

See, we cannot express this limit-of-a-quotient as a-quotient-of-the-limits, then the derivative is not a quotient.
More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__.

Can we perform operation without relying on this assumption?
The answer is yes! Using the chain rule.

Lagrange's Notation
-------------------
Also cal prime notation

Newton's Notation
-------------------


.. note::
   These notes are inspired from `this page <https://byjus.com/maths/ordinary-differential-equations/>`__