Lösung 1.2:1f

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In this case, we have a slightly more complicated expression, but if we focus on the expression's outer form, we have essentially "something divided by
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In this case, we have a slightly more complicated expression, but if we focus on the expression's outer form, we have essentially "something divided by <math>\sin x</math>". As a first step, we therefore use the quotient rule,
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<math>\sin x</math>
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". As a first step, we therefore use the quotient rule,
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{{Displayed math||<math>\Bigl(\frac{x\ln x}{\sin x}\Bigr)' = \frac{(x\ln x)'\cdot \sin x - x\ln x\cdot (\sin x)'}{(\sin x)^2}\,\textrm{.}</math>}}
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<math>\left( \frac{x\ln x}{\sin x} \right)^{\prime }=\frac{\left( x\ln x \right)^{\prime }\centerdot \sin x-x\ln x\centerdot \left( \sin x \right)^{\prime }}{\left( \sin x \right)^{2}}</math>
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We can, in turn, differentiate the expression <math>x\ln x</math> by using the product rule,
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We can, in turn, differentiate the expression
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<math>x\ln x</math>
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by using the product rule:
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<math>\begin{align}
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& \left( x\ln x \right)^{\prime }=\left( x \right)^{\prime }\ln x+x\left( \ln x \right)^{\prime } \\
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& \\
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& =1\centerdot \ln x+x\centerdot \frac{1}{x}=\ln x+1 \\
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\end{align}</math>
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{{Displayed math||<math>\begin{align}
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(x\ln x)'
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&= (x)'\ln x + x\,(\ln x)'\\[5pt]
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&= 1\cdot\ln x + x\cdot\frac{1}{x}\\[5pt]
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&= \ln x+1\,\textrm{.}
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\end{align}</math>}}
All in all, we thus obtain
All in all, we thus obtain
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{{Displayed math||<math>\begin{align}
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<math>\begin{align}
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\Bigl(\frac{x\ln x}{\sin x}\Bigr)'
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& \left( \frac{x\ln x}{\sin x} \right)^{\prime }=\frac{\left( \ln x+1 \right)\centerdot \sin x-x\ln x\centerdot \cos x}{\left( \sin x \right)^{2}} \\
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&= \frac{(\ln x+1)\cdot\sin x - x\ln x\cdot \cos x}{(\sin x)^2}\\[5pt]
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& \\
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&= \frac{\ln x+1}{\sin x}-\frac{x\ln x\cos x}{\sin^2\!x}\,\textrm{.}
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& =\frac{\ln x+1}{\sin x}-\frac{x\ln x\cos x}{\sin ^{2}x} \\
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\end{align}</math>}}
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\end{align}</math>
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Version vom 14:22, 14. Okt. 2008

In this case, we have a slightly more complicated expression, but if we focus on the expression's outer form, we have essentially "something divided by \displaystyle \sin x". As a first step, we therefore use the quotient rule,

\displaystyle \Bigl(\frac{x\ln x}{\sin x}\Bigr)' = \frac{(x\ln x)'\cdot \sin x - x\ln x\cdot (\sin x)'}{(\sin x)^2}\,\textrm{.}

We can, in turn, differentiate the expression \displaystyle x\ln x by using the product rule,

\displaystyle \begin{align}

(x\ln x)' &= (x)'\ln x + x\,(\ln x)'\\[5pt] &= 1\cdot\ln x + x\cdot\frac{1}{x}\\[5pt] &= \ln x+1\,\textrm{.} \end{align}

All in all, we thus obtain

\displaystyle \begin{align}

\Bigl(\frac{x\ln x}{\sin x}\Bigr)' &= \frac{(\ln x+1)\cdot\sin x - x\ln x\cdot \cos x}{(\sin x)^2}\\[5pt] &= \frac{\ln x+1}{\sin x}-\frac{x\ln x\cos x}{\sin^2\!x}\,\textrm{.} \end{align}

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