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Lösung 1.2:4a

Aus Online Mathematik Brückenkurs 2

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We are to differentiate the expression two times, so we start by differentiating once. The quotient rule gives
We are to differentiate the expression two times, so we start by differentiating once. The quotient rule gives
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{{Displayed math||<math>\begin{align}
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\frac{d}{dx}\,\frac{x}{\sqrt{1-x^2}}
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&= {}\rlap{\frac{(x)'\sqrt{1-x^2}-x\bigl(\sqrt{1-x^2}\bigr)'}{\bigl(\sqrt{1-x^2}\bigr)^2}}\phantom{\frac{\sqrt{1-x^2}-x\cdot\dfrac{1}{2\sqrt{1-x^2}}\cdot\bigl(1-x^2\bigr)'}{1-x^2}}\\[5pt]
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&= \frac{1\cdot\sqrt{1-x^2}-x\bigl(\sqrt{1-x^2}\bigr)'}{1-x^2}\,\textrm{.}
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\end{align}</math>}}
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<math>\begin{align}
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We determine the derivative <math>\bigl(\sqrt{1-x^2}\bigr)'</math> by using the chain rule
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& \frac{d}{dx}\frac{x}{\sqrt{1-x^{2}}}=\frac{\left( x \right)^{\prime }\sqrt{1-x^{2}}-x\left( \sqrt{1-x^{2}} \right)^{\prime }}{\left( \sqrt{1-x^{2}} \right)^{2}} \\
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& \\
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& =\frac{1\centerdot \sqrt{1-x^{2}}-x\left( \sqrt{1-x^{2}} \right)^{\prime }}{1-x^{2}} \\
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\end{align}</math>
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{{Displayed math||<math>\begin{align}
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\phantom{\frac{d}{dx}\,\frac{x}{\sqrt{1-x^2}}}{}
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&= \frac{\sqrt{1-x^2}-x\cdot\dfrac{1}{2\sqrt{1-x^2}}\cdot\bigl(1-x^2\bigr)'}{1-x^2}\\[5pt]
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&= \frac{\sqrt{1-x^2}-x\cdot\dfrac{1}{2\sqrt{1-x^2}}\cdot (-2x)}{1-x^2}\,\textrm{.}
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\end{align}</math>}}
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We determine the derivative
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We simplify the result as far as possible, so as to make the second differentiation easier,
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<math>\left( \sqrt{1-x^{2}} \right)^{\prime }</math>
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by using the chain rule
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{{Displayed math||<math>\begin{align}
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\phantom{\frac{d}{dx}\,\frac{x}{\sqrt{1-x^2}}}{}
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&= {}\rlap{\frac{\sqrt{1-x^2} + \dfrac{x^2}{\sqrt{1-x^2}}}{1-x^2}}\phantom{\frac{\sqrt{1-x^2}-x\cdot\dfrac{1}{2\sqrt{1-x^2}}\cdot\bigl(1-x^2\bigr)'}{1-x^2}}\\[5pt]
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&= \frac{\dfrac{\bigl(\sqrt{1-x^2}\bigr)^2}{\sqrt{1-x^2}}+\dfrac{x^2}{\sqrt{1-x^2}}}{1-x^2}\\[5pt]
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&= \frac{\dfrac{1-x^2+x^2}{\sqrt{1-x^2}}}{1-x^2}\\[5pt]
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&= \frac{1}{(1-x^2)^{3/2}}\,\textrm{.}
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\end{align}</math>}}
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<math>\begin{align}
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The second derivative is
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& =\frac{\sqrt{1-x^{2}}-x\centerdot \frac{1}{2\sqrt{1-x^{2}}}\centerdot \left( 1-x^{2} \right)^{\prime }}{1-x^{2}} \\
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& \\
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& =\frac{\sqrt{1-x^{2}}-x\centerdot \frac{1}{2\sqrt{1-x^{2}}}\centerdot \left( -2x \right)}{1-x^{2}} \\
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\end{align}</math>
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{{Displayed math||<math>\begin{align}
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We simplify the result as far as possible, so as to make the second differentiation easier:
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\frac{d^2}{dx^2}\,\frac{x}{\sqrt{1-x^2}}
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&= \frac{d}{dx}\,\frac{1}{(1-x^2)^{3/2}}\\[5pt]
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&= \frac{d}{dx}\,\bigl(1-x^2\bigr)^{-3/2}\\[5pt]
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<math>\begin{align}
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&= -\tfrac{3}{2}\bigl(1-x^2\bigr)^{-3/2-1}\cdot\bigl(1-x^2\bigr)'\\[5pt]
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& =\frac{\sqrt{1-x^{2}}+\frac{x^{2}}{\sqrt{1-x^{2}}}}{1-x^{2}} \\
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&= -\tfrac{3}{2}\bigl(1-x^2\bigr)^{-5/2}\cdot (-2x)\\[5pt]
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& \\
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&= 3x\bigl(1-x^2\bigr)^{-5/2}\\[5pt]
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& =\frac{\frac{\left( \sqrt{1-x^{2}} \right)^{2}}{\sqrt{1-x^{2}}}+\frac{x^{2}}{\sqrt{1-x^{2}}}}{1-x^{2}} \\
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&= \frac{3x}{\bigl(1-x^2\bigr)^{5/2}}\,\textrm{.}
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& \\
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\end{align}</math>}}
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& =\frac{\frac{1-x^{2}+x^{2}}{\sqrt{1-x^{2}}}}{1-x^{2}} \\
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& \\
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& =\frac{1}{\left( 1-x^{2} \right)^{{3}/{2}\;}} \\
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\end{align}</math>
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The second derivative is:
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<math>\begin{align}
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& \frac{d^{^{2}}}{dx^{^{2}}}\frac{x}{\sqrt{1-x^{2}}}=\frac{d}{dx}\frac{1}{\left( 1-x^{2} \right)^{{3}/{2}\;}} \\
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& \\
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& =\frac{d}{dx}\left( 1-x^{2} \right)^{-{3}/{2}\;}=-\frac{3}{2}\left( 1-x^{2} \right)^{-\frac{3}{2}-1}\centerdot \left( 1-x^{2} \right)^{\prime } \\
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& \\
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& =-\frac{3}{2}\left( 1-x^{2} \right)^{{-5}/{2}\;}\centerdot \left( -2x \right)=3x\left( 1-x^{2} \right)^{{-5}/{2}\;} \\
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& \\
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& =\frac{3x}{\left( 1-x^{2} \right)^{{5}/{2}\;}} \\
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\end{align}</math>
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Version vom 13:47, 15. Okt. 2008

We are to differentiate the expression two times, so we start by differentiating once. The quotient rule gives

ddxx1x2=1x22(x)1x2x1x2=1x211x2x1x2.

We determine the derivative 1x2  by using the chain rule

=1x21x2x121x21x2=1x21x2x121x2(2x).

We simplify the result as far as possible, so as to make the second differentiation easier,

=1x21x2+x21x2=1x21x21x22+x21x2=1x21x21x2+x2=1(1x2)32.

The second derivative is

d2dx2x1x2=ddx1(1x2)32=ddx1x232=231x23211x2=231x252(2x)=3x1x252=3x1x252.