The slope of a line is defined to be the rise divided by the run. For a curved line, however, the slope has to be defined at a point. Lines (curved or straight, but not infinitely steep) can always be thought of as functions of a single variable. We call the slope of a line evaluated at any given point its derivative, and call the process of finding that slope taking the derivative of the function.
Later we'll say a few words about multivariate (vector) differential calculus, but that is mostly beyond the scope of this course.
The definition of the derivative of a function is:
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(86) |
First, note that:
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(87) |
Second, differentiation is linear. That is:
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(88) |
Third, suppose that
(the product of two functions). Then
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(89) |
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(90) |
We can easily and directly compute the derivative of a mononomial:
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(91) |
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(92) |
Again it is beyond the scope of this short review to completely
rederive all of the results of a calculus class, but from what has been
presented already one can see how one can systematically proceed. We
conclude, therefore, with a simple table of useful derivatives and
results in summary (including those above):
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(93) |
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(94) |
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(95) |
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(96) |
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(97) |
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(98) |
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(99) |
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(100) |
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(101) |
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(102) |
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(103) |
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(104) |
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(105) |
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(106) |