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Differentiation

Differentiation is also called taking the derivative, and the result of differentiation is the derivative. The calculus that concerns taking the derivative (and its implications) is called differential calculus. Taking the derivative (or differentiation) is an operation that take a function for its input and produces a function as its output.

{$$ {\large \left. \begin{matrix} x^2 + 5 \cr x^2 + 3 \cr x^2 - 17 \cr x^2 + \sqrt 2 \end{matrix} \right\} \Rightarrow {\Large \boxed{\;\operatorname{D}\;}} \Rightarrow 2x } $$}

While it is possible to use particular values to find a value of a derivative directly, this is seldom done except as demonstrations for beginning students. In general, the derivative is derived of a function, results in a function, and particular values are found by substitution (plugging in) to the resulting function.

As might be inferred from the word differentiation, the derivative involves a difference. The derivative is about a difference: a difference between here and there, now and then, high and low. Generally we say it is the instantaneous rate of change, but neither the instantaneous nor the change, should imply that the difference is necessarily a difference in time. Of course the first practical problems in calculus often are mechanics problem where the difference is in time, but this not the general case.

Δ is the Greek letter delta. In mathematics &Delta often represents a difference or a change in value.

In calculus in particular Δ represents a relatively small but finite change or difference in the value of a variable.

{$ \Delta y $} means a small difference in y.

Geometrically, finding the derivative of a function means finding a formula for the slope of the line that is tangent to the graph of function at any given point on the graph of the function. To illustate what this means, the particular example in Figure 1 is toured.


Figure 1 Taking the derivative is finding the slope of the tangent.

A Tour of {$ \displaystyle {1 \over x}$}


Graph:

Sources:

  1. Portrait of Isaac Newton by Sir Godfrey Kneller Wiki Commons

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Category: Math


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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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August 05, 2017

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