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Mathematics-Online course: Prepcourse Mathematics - Analysis - Differential Calculus

Derivative

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The function $ f$ is differentiable in the point $ a$ if the limit

$\displaystyle f'(a)= \lim_{h\rightarrow0} \frac{f(a+h)-f(a)}{h} $

exists. This limit is called derivative.

\includegraphics[width=0.6\linewidth]{a_ableitung_bild_1.eps}

The geometrical interpretation of differentiability is that the slopes of the secants converge to the slope of the tangent given by

$\displaystyle y=f(a)+f'(a)(x-a). $

One also writes $ f'(x)=\displaystyle\frac{d}{dx}f(x)$ and denotes higher derivatives by $ f'',f''',\ldots$ resp. $ f^{(2)},f^{(3)},\ldots$.

Derivative of ALT=:

$\displaystyle f'(x) = \lim_{h\rightarrow0} \frac{(x+h)^2-x^2}{h}=\lim_{h\rightarrow0} \frac{2xh+h^2}{h}=\lim_{h\rightarrow0} (2x+h)=2x\,. $

Its second derivative $ f''(x) = 2$ is constant.

Generally, with the binomial theorem it is for any given monomial $ f(x) = x^n$, $ n\in\mathbb{N}$,

$\displaystyle f'(x) = \lim_{h \rightarrow 0} \frac{(x+h)^n - x^n}{h} = \lim_{h ... ...begin{array}{c} n \\ 1 \end{array} \right) x^{n-1}h + O(h^2)}{h} = nx^{n-1}\,. $

(Authors: Höllig/Kopf/Abele)
The derivative of ALT= can be determined with the help of the addition theorem.
From

ALT=

it follows with ALT= that the difference quotient is

ALT=

Since

ALT=

the right-hand side of the equation strives towards ALT= for ALT= .

(Authors: Höllig/Kopf/Abele)
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  automatically generated 10/23/2009