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Derivative

lastmod 2018-09-04created 2014-09-0328 edits

Differentiation (Derivative) is the calculation process of finding a function's instantaneous rate of change.

A derivative means the instantaneous rate of change of a continuous, ongoing change. Therefore, differentiation is impossible for non-continuous changes or at singular points such as cusps. On a function's graph, the derivative can be represented as the tangent line at a particular point on the function's curve.

For the derivatives of various functions, refer to the table of derivatives1.

Table of derivatives

Various differentiation results and formulas are summarized below.

  • \({d \over dx} x^c = cx^{c-1}\)
  • \({d \over dx} x = x^0 = 1\)
  • \({d \over dx} c = 0\)
  • \({d \over dx} \sqrt{x} = {1 \over 2 \sqrt{x}}\)
  • \({d \over dx} \left({1 \over x}\right) = -{1 \over x^2}\)

Derivative function

As seen in the overview above, differentiating the function

\[ f(x) = k x^2 + c \] produces a function with a new relation,

\[ f'(x)=2k x^{2-1} \] .

A function produced by differentiating the original function in this way is called the derivative. In general, the derivative of a function defined by a polynomial equation can be obtained by differentiation. That is, the derivative f'(x) of a function f(x) can be written as follows.

\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \]

Solution

For example,

\[ f(x) = x^2 - 4\ \] has the derivative

\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \]

\[ =\lim_{\Delta x \to 0} \frac{ {(x + \Delta x)}^2 - 4 - (x^2 - 4)}{\Delta x} \]

\[ =\lim_{\Delta x \to 0} \frac{ x^2 + 2 x \Delta x + {\Delta x}^2 - 4 - x^2 + 4}{\Delta x} \]

\[ =\lim_{\Delta x \to 0} \frac{ 2 x \Delta x + {\Delta x}^2 }{\Delta x} \]

\[ =\lim_{\Delta x \to 0} 2 x + \Delta x \]

\[ f'(x) = 2x\ \] .

Partial derivative

Partial differentiation (partial derivative; Partial derivative) is differentiating a multivariable function with respect to a particular variable while treating the remaining variables as constants. This concept is used importantly in vector calculus.

∂ is a rounded d, a variant form of the letter delta δ, and is read as "partial".

Differential

Differential A differential (微分素; Differential) is an infinitesimal value representing an infinitely small change in a function, written as \(\mathrm d y\). Symbols such as \(\Delta x\) and \(\delta x\) are usually used to denote a change in a function, but \(\mathrm d x\) is distinguished from them in that it means an infinitely small value.

Terminology

Delta (d; △)
A symbol representing a difference in values.
Average rate of change.
See: Differentiation (1) - Slope and average rate of change - What is the difference? 3
Rate of change on the closed interval [a,b] (= △y/△x) ⇔ slope of segment AB.
Instantaneous rate of change.
See: Differentiation (2) - Instantaneous rate of change 4
Rate of change at x = a (= f'(a)) ⇔ slope of the tangent line.
Differential coefficient and Derivative.
See: Differentiation (3) Differential coefficient, derivative 5
The function f'(x) obtained by differentiating the function y=f(x). Generally also called the derivative coefficient or differential coefficient of f(x).

See also

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References

  1. Wikipedia_list_of_differentiation_rules_ko.pdf ↩

  2. Partial_Differentiation.pdf ↩

  3. Gradient-and-average_rate_of_change.pdf ↩

  4. Instantaneous_rate_of_change.pdf ↩

  5. Derivative-and-differential_coefficient.pdf ↩

  6. Wikipedia_derivative_ko.pdf ↩

  7. What_is_differentiate.zip ↩

  8. Differential_coefficient_and_rate_of_change.pdf ↩

  9. Darkpgmr_-_Derivative_and_Integral.pdf ↩

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