See also: derivative - I didn't see this until after I had created this node.


In calculus differentiation is the process of writing explicitly how an equation differs or changes its value when the variable (or one of the variables) change.

A simple example is the one of the clock. First imagine a old-time clock made of gears and such. One imagine that as the minute hand is moved, the second hand (the one that counts seconds) will have to move 60 times as fast in terms of the angle traveled. The hour hand, on the other hand (pun intended) will only move 1/60th of the angle that the minute hand moves. While only a simple example, differentiation is just this exact thing applied do a host of different problems.

Along with the other fundamental concepts of calculus, differentiation was codified by Isaac Newton. Differentiation is considerably easier to do than integration because there is a formula to follow to achieve the result and there are now in general a set of rules for how to differentiate most functions.

The fundamental definition of the derivative is: df(x)/(dx) = limit as h goes to zero of (f(x+h) - f(x))/h) All the other 'rules' can be derived from this fundamental concept. The short-hand notation is either f with a dot over it or f'

Differentiation Rules:
  • Constants: Constants don't change, the derivative is zero.
  • Powers: dxn/(dx) = n*xn-1
  • Linear Combination: (f(x) + g(x))' = f'(x) + g'(x)
  • Product rule: (f(x)g(x))' = g(x)f'(x) + f(x)g'(x)
  • Reciprocal rule: (1/f(x))' = -f'(x)/(f(x)^2)
  • The Quotient rule: (f(x)/g(x))' = g(x)f'(x) - f(x)g'(x) * 1/g(x)^2

    There are quite a few other rules but I think all of them can be derived from these . . .

  • Dif`fer*en`ti*a"tion (?), n.


    The act of differentiating.

    Further investigation of the Sanskrit may lead to differentiation of the meaning of such of these roots as are real roots. J. Peile.

    2. Logic

    The act of distinguishing or describing a thing, by giving its different, or specific difference; exact definition or determination.

    3. Biol.

    The gradual formation or production of organs or parts by a process of evolution or development, as when the seed develops the root and the stem, the initial stem develops the leaf, branches, and flower buds; or in animal life, when the germ evolves the digestive and other organs and members, or when the animals as they advance in organization acquire special organs for specific purposes.

    4. Metaph.

    The supposed act or tendency in being of every kind, whether organic or inorganic, to assume or produce a more complex structure or functions.


    © Webster 1913.

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