41 lines
1.5 KiB
ReStructuredText
41 lines
1.5 KiB
ReStructuredText
Differentiation Notations
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================================
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Leibniz's Notation
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-------------------
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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}`.
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The Leibniz's notation expresses the derivative of :math:`f` as :math:`\frac{dy}{dx}` with :math:`y=f(x)`.
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More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__.
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.. note::
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The following :math:`\frac{dx}{dt}` means :math:`x` is a function of :math:`t` such as :math:`x=f(t)`.
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See explanations `here <https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-4/v/differentiating-related-functions-intro>`__.
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:math:`\frac{d}{dx}` is an operator not a quotient!
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Although it behaves like a quotient. In fact it is a limit:
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.. math::
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\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}
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See, we cannot express this limit-of-a-quotient as a-quotient-of-the-limits, then the derivative is not a quotient.
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More details `here <https://www.me.psu.edu/cimbala/me420/Homework/dydx_quotient_article.html>`__.
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Can we perform operation without relying on this assumption?
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The answer is yes! Using the chain rule.
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Lagrange's Notation
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-------------------
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Also cal prime notation
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Newton's Notation
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-------------------
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.. note::
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These notes are inspired from `this page <https://byjus.com/maths/ordinary-differential-equations/>`__
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