Lösung 1.1:2f

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We can rewrite the function using a trigonometric addition formula:
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<center> [[Image:1_1_2f.gif]] </center>
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<math>f\left( x \right)=\cos \left( x+\frac{\pi }{3} \right)=\cos x\centerdot \cos \frac{\pi }{3}-\sin x\centerdot \sin \frac{\pi }{3}</math>
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If we now differentiate this expression,
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<math>\cos \frac{\pi }{3}</math>
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and
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<math>\sin \frac{\pi }{3}</math>
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are constants and we obtain
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<math>\begin{align}
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& {f}'\left( x \right)=\frac{d}{dx}\left( \cos x\centerdot \cos \frac{\pi }{3}-\sin x\centerdot \sin \frac{\pi }{3} \right) \\
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& =\cos \frac{\pi }{3}\centerdot \frac{d}{dx}\cos x-\sin \frac{\pi }{3}\centerdot \frac{d}{dx}\sin x \\
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& =\cos \frac{\pi }{3}\centerdot \left( -\sin x \right)-\sin \frac{\pi }{3}\centerdot \cos x \\
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\end{align}</math>
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If we then use the addition formula in reverse, this gives
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<math>\begin{align}
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& {f}'\left( x \right)=-\left( \sin x\centerdot \cos \frac{\pi }{3}+\cos x\centerdot \sin \frac{\pi }{3} \right) \\
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& =-\sin \left( x+\frac{\pi }{3} \right) \\
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\end{align}</math>
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NOTE: In the next section, we will go through differentiation rules which make it possible to differentiate the expression directly without rewriting in this way.

Version vom 11:49, 10. Okt. 2008

We can rewrite the function using a trigonometric addition formula:


\displaystyle f\left( x \right)=\cos \left( x+\frac{\pi }{3} \right)=\cos x\centerdot \cos \frac{\pi }{3}-\sin x\centerdot \sin \frac{\pi }{3}


If we now differentiate this expression, \displaystyle \cos \frac{\pi }{3} and \displaystyle \sin \frac{\pi }{3} are constants and we obtain


\displaystyle \begin{align} & {f}'\left( x \right)=\frac{d}{dx}\left( \cos x\centerdot \cos \frac{\pi }{3}-\sin x\centerdot \sin \frac{\pi }{3} \right) \\ & =\cos \frac{\pi }{3}\centerdot \frac{d}{dx}\cos x-\sin \frac{\pi }{3}\centerdot \frac{d}{dx}\sin x \\ & =\cos \frac{\pi }{3}\centerdot \left( -\sin x \right)-\sin \frac{\pi }{3}\centerdot \cos x \\ \end{align}


If we then use the addition formula in reverse, this gives


\displaystyle \begin{align} & {f}'\left( x \right)=-\left( \sin x\centerdot \cos \frac{\pi }{3}+\cos x\centerdot \sin \frac{\pi }{3} \right) \\ & =-\sin \left( x+\frac{\pi }{3} \right) \\ \end{align}

NOTE: In the next section, we will go through differentiation rules which make it possible to differentiate the expression directly without rewriting in this way.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_1.1:2f“