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Mathematics-Online course: Preparatory Course Mathematics - Analysis - Differential Calculus

Derivative

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A function $ f$ is differentiable at the point $ a$ if the limit

$\displaystyle f^\prime(a)= \lim_{h\to0} \frac{f(a+h)-f(a)}{h} $

exists. This limit is called the derivative of $ f$ at $ a$.

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

Geometrically, differentiability means that the slopes of the secants converge to the slope of the tangent given by

$\displaystyle y=f(a)+f^\prime(a)(x-a) \,. $

We also write

$\displaystyle f^{\prime}(x)=\frac{d}{dx}f(x)=\frac{dy}{dx} $

with $ y=f(x)$. This notation symbolizes the limit $ \Delta x\to0$ for the difference quotient.

Higher derivatives are denoted by $ f^{\prime\prime},f^{\prime\prime\prime},\ldots$ or $ f^{(2)},f^{(3)},\ldots$, respectively.

We say that a function $ f$ is differentiable on a set $ D$ if $ f^\prime(x)$ exists for all $ x\in D$.

According to the definition, the derivative of the function $ f(x)=x^2$ is given by

$\displaystyle f^\prime(x) = \lim_{h\to0} \frac{(x+h)^2-x^2}{h}=\lim_{h\to0} \frac{2xh+h^2}{h}=\lim_{h\to0} (2x+h)=2x\,. $

The seconed derivative is constant: $ f^{\prime\prime}(x) = 2$.

In general, with the aid of the binomial formula, we can compute the derivative of an arbitrary monomial $ f(x) = x^n$:

$\displaystyle f^\prime(x) = \lim_{h \to0} \frac{(x+h)^n - x^n}{h} = \lim_{h \to0} \frac{\binom{n}{1} x^{n-1}h + O(h^2)}{h} = nx^{n-1}\,, $

with $ O(h^2)$ denoting terms of order $ h^2$.
The derivative of $ f(x)=\sin x $ can be determined with the aid of the addition theorem. From

$\displaystyle \sin(t\pm h/2)=\sin t\cos(h/2) \pm \cos t \sin(h/2) $

with $ t=x+h/2$, it follows that the difference quotient equals

$\displaystyle \frac{\sin(x+h)-\sin x }{h} = \frac{\sin\big((x+h/2)+h/2\big)-\sin\big((x+h/2)-h/2\big)}{h} = \frac{2\cos(x+h/2)\sin(h/2)}{h}\,. $

Since

$\displaystyle \lim_{h \to0} \frac{2\sin(h/2)}{h}= \lim_{h \to0} \frac{\sin(h/2)}{h/2}=1\,, $

the right-hand side tends to $ \cos x $ for $ h\to0$ .
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  automatically generated 1/9/2017