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

\displaystyle \begin{align}

\frac{d}{dx}\,\frac{x}{\sqrt{1-x^2}} &= {}\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] &= \frac{1\cdot\sqrt{1-x^2}-x\bigl(\sqrt{1-x^2}\bigr)'}{1-x^2}\,\textrm{.} \end{align}

We determine the derivative \displaystyle \bigl(\sqrt{1-x^2}\bigr)' by using the chain rule

\displaystyle \begin{align}

\phantom{\frac{d}{dx}\,\frac{x}{\sqrt{1-x^2}}}{} &= \frac{\sqrt{1-x^2}-x\cdot\dfrac{1}{2\sqrt{1-x^2}}\cdot\bigl(1-x^2\bigr)'}{1-x^2}\\[5pt] &= \frac{\sqrt{1-x^2}-x\cdot\dfrac{1}{2\sqrt{1-x^2}}\cdot (-2x)}{1-x^2}\,\textrm{.} \end{align}

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

\displaystyle \begin{align}

\phantom{\frac{d}{dx}\,\frac{x}{\sqrt{1-x^2}}}{} &= {}\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] &= \frac{\dfrac{\bigl(\sqrt{1-x^2}\bigr)^2}{\sqrt{1-x^2}}+\dfrac{x^2}{\sqrt{1-x^2}}}{1-x^2}\\[5pt] &= \frac{\dfrac{1-x^2+x^2}{\sqrt{1-x^2}}}{1-x^2}\\[5pt] &= \frac{1}{(1-x^2)^{3/2}}\,\textrm{.} \end{align}

The second derivative is

\displaystyle \begin{align}

\frac{d^2}{dx^2}\,\frac{x}{\sqrt{1-x^2}} &= \frac{d}{dx}\,\frac{1}{(1-x^2)^{3/2}}\\[5pt] &= \frac{d}{dx}\,\bigl(1-x^2\bigr)^{-3/2}\\[5pt] &= -\tfrac{3}{2}\bigl(1-x^2\bigr)^{-3/2-1}\cdot\bigl(1-x^2\bigr)'\\[5pt] &= -\tfrac{3}{2}\bigl(1-x^2\bigr)^{-5/2}\cdot (-2x)\\[5pt] &= 3x\bigl(1-x^2\bigr)^{-5/2}\\[5pt] &= \frac{3x}{\bigl(1-x^2\bigr)^{5/2}}\,\textrm{.} \end{align}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_1.2:4a“