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Calculus Derivations

  1. Feb 17, 2016 #1
  2. jcsd
  3. Feb 17, 2016 #2


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    I think any introductory calculus text will cover these rules in detail.
    General properties of derivatives can be demonstrated using the definition of the derivative as a limit.
    ##f'(x) = \lim_{h \to 0 } \frac{f(x+h) - f(x) }{h}##
    For example, the product rule is:
    ##\frac{d}{dx}(fg) = \lim_{h \to 0 } \frac{f(x+h)g(x+h) - f(x)g(x) }{h}##
    Adding zero in the form of ##f(x+h)g(x)-f(x+h)g(x)##:
    ##\qquad = \lim_{h \to 0 } \frac{f(x+h)g(x+h) - f(x)g(x) +f(x+h)g(x)-f(x+h)g(x) }{h}\\
    \qquad = \lim_{h \to 0 } \frac{[f(x+h)g(x+h) -f(x+h)g(x)]+[f(x+h)g(x) - f(x)g(x)] }{h}\\
    \qquad = \lim_{h \to 0 } f(x+h)\frac{g(x+h) -g(x)}{h}+g(x)\frac{f(x+h)- f(x) }{h}##
    Applying the limit, this will be:
    ##\frac{d}{dx}(fg) = f(x)\frac{dg}{dx} + g(x)\frac{df}{dx}.##
    The integral rules are often demonstrated as the inverse of the derivative...i.e. once you know all the derivative rules, apply the derivatives to the integrals to verify them for yourself.
  4. Feb 17, 2016 #3


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    Just about any textbook on differential and integral calculus will show how to derive these formulas, starting with No. (1) on the Table of Integrals. Outside of a few basic techniques and the integrals of the elementary functions, the rest of the formulas on these tables are just applications to more complex combinations of the elementary functions. The derivation of the formulas for the derivatives of the various elementary functions can be obtained from the definition of the derivative.

    The tables themselves are compiled so that someone can quickly determine the integral of one of the forms contained there without going through all of the scut work that is expected of a student. Of course, with computers now doing symbolic algebra and calculus, such tables are no longer needed as much as they once were.
  5. Feb 23, 2016 #4
    Thanks, sorry for the delayed response I learn a variety of things throughout the week.

    I started reading an analysis book and managed to prove the simple power law for both differentiating and integrating. The integral derivation used the sum of squares relationship to simplify which was something to prove in itself.
    Also derived the derivatives of sinx, cosx, 1/x and e^x, which all used niche identities.

    I can assume that the rest of the integrals on this table, like with 1/x = lnx that I am currently trying to do, all require re-expressing sums of sequences and required identities. I plan on filling a notebook of these derivations, its going to take a long time.
  6. Feb 23, 2016 #5


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    Whatever floats your boat.

    The important thing is not to derive every single integration formula from scratch, but to understand the different integration techniques which are used to simplify certain integrals, like u-substitution or integration by parts.

    While the tables of indefinite integrals you linked to are fairly comprehensive, there are entire books filled with such formulae, which have been derived over many years by many different mathematicians. One such reference, which includes definite and indefinite integrals of various types, is this one:


    There's almost 1200 pages of integrals, derivatives, series, etc. in this book, which is one of the more comprehensive tomes available on the subject.
    Last edited by a moderator: May 7, 2017
  7. Feb 23, 2016 #6
    Ah, right, yes.
    In that case I'll only do a few more and then start doing calculus for physics. Thanks for putting the wind in my sails.
  8. Feb 23, 2016 #7


    Staff: Mentor

    Last edited by a moderator: May 7, 2017
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