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Derivative (calculus) (view source)
Revision as of 21:44, November 29, 2009
, 21:44, November 29, 2009Restored "Higher order derivatives" section from page history, with some abridgements
For example, <math>\frac{d}{dx} ( 2x ) =\lim_{h \to 0}\frac{2(x+h)-2(x)}{h} = \lim_{h \to 0}\frac{2x+2h-2x}{h} = \lim_{h \to 0}\frac{2h}{h} = \lim_{h \to 0}\ 2 = 2</math>
For example, <math>\frac{d}{dx} ( 2x ) =\lim_{h \to 0}\frac{2(x+h)-2(x)}{h} = \lim_{h \to 0}\frac{2x+2h-2x}{h} = \lim_{h \to 0}\frac{2h}{h} = \lim_{h \to 0}\ 2 = 2</math>
In general, <tt>f'(mx) = m</tt>; that is, the derivative of any line is equal to its slope.
In general, <tt>f'(mx) = m</tt>; that is, the derivative of any line is equal to its slope.
===Higher order derivatives===
A higher order derivative is obtained by repeatedly differentiating a function. Thus, the second derivative of x, or <math>\frac{d^{2}y}{dx^{2}}</math>, is <math>\frac{d}{dx}\left(\frac{dy}{dx}\right)</math>
Similarly,
:<math>\frac{d^{3}y}{dx^{3}}=\frac{d}{dx}\left(\frac{d^{2}y}{dx^{2}}\right)=\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right)</math>
and so forth.
A common alternative notation is <math>f''(x)</math>, <math>f'''(x)</math>, and <math>f^{(n)}(x)</math>
==Uses==
==Uses==