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jancraats:hoofdstuk_20 [2012/10/05 12:13] – jerryjancraats:hoofdstuk_20 [2022/09/01 11:44] (current) – external edit 127.0.0.1
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-[[jancraats/hoofdstuk_20/pagina_176 | Pagina 176 - Opgaves van 20.1 tot 20.9]]+<note tip> 
 +Let op de **\rightarrow** (latex-opmaak) bij differentieren, gebruik geen gelijk-aan teken. Functies zijn niet gelijk aan elkaar, ze worden getransformeerd. 
 +</note>
  
-====== 20.10 ====== +  * [[jancraats/hoofdstuk_20/pagina_176 | Pagina 176 - Opgaves van 20.1 tot 20.9]] 
- +  * [[jancraats/hoofdstuk_20/pagina_178 | Pagina 178 - Opgaves van 20.10 tot 20.18]] 
-**e** +  * [[jancraats/hoofdstuk_20/pagina_180 | Pagina 180 - Opgaves van 20.19 tot 20.31]]
- +
-$$ +
-\begin{split}  +
- +
-(x - x^4) ^ {-2} \\ +
--2 (x - x^4) ^ {-3} \cdot 1 - 4 x^3 \\ +
--2(x - x^4) ^ {-3} +
- +
-\end{split} +
-$$ +
- +
-Behalve de kettingregel gebruik je ook de somregel! +
- +
-====== 20.11 ====== +
- +
-**a** +
-  +
-$$ +
-\begin{split}  +
-  +
-((2x-3)^5)' &\rightarrow 5 ( 2x - 3 )^4 \cdot 2 \\ +
-&\rightarrow 10 ( 2x - 3 )^4 +
-  +
-\end{split} +
-$$ +
-  +
-**b** +
-  +
-$$ +
-\begin{split}  +
-  +
-((x^2 + 5)^{-1})' &\rightarrow -1 (x^2 + 5)^{-2} \cdot 2x \\ +
-&\rightarrow -2x(x^2+5)^{-2} +
-  +
-\end{split} +
-$$ +
-  +
-**c** +
-  +
-$$ +
-\begin{split}  +
-  +
-((\sqrt{3x - 4}))' &\rightarrow \frac{1}{2} (3x - 4)^{ - \frac{1}{2} } \cdot 3 \\ +
-&\rightarrow \frac{3}{2} (3x - 4)^{ - \frac{1}{2} } +
-  +
-\end{split} +
-$$ +
-  +
-**d** +
-  +
-$$ +
-\begin{split}  +
- +
-((\sqrt{x^2 + x}))' &\rightarrow \frac{1}{2} (x^2 + x)^{ - \frac{1}{2} } \cdot ( 2x + 1 ) \\ +
-&\rightarrow \frac{1}{2} ( 2x + 1 )( x^2 + x )^{ - \frac{1}{2} } +
-  +
-\end{split} +
-$$ +
-  +
-**e** +
-  +
-$$ +
-\begin{split}  +
-  +
-((x + 4x^3)^{-3})' &\rightarrow -3 ( x + 4x^3 )^{-4} \cdot ( 1 + 12x^2 ) \\ +
-&\rightarrow -3 ( 1 + 12x^2 )( x + 4x^3 )^{ -4 } +
-  +
-\end{split} +
-$$ +
- +
-====== 20.12 ====== +
-  +
-**a** +
-  +
-$$ +
-\begin{split}  +
-  +
-(\sqrt{ 1 + x + x^2 })' &\rightarrow \frac{1}{2} ( 1 + x + x^2 )^{ - \frac{1}{2} } \cdot ( 1 + 2x ) \\ +
-&\rightarrow \frac{1}{2} ( 1 + 2x )( 1 + x + x^2 )^{ - \frac{1}{2} } +
-  +
-\end{split} +
-$$ +
-  +
-**b** +
-  +
-$$ +
-\begin{split}  +
- +
-(\sqrt[3]{ 1 + x + x^2 })' &\rightarrow \frac{1}{3} ( 1 + x + x^2 )^{ - \frac{2}{3} } \cdot ( 1 + 2x ) \\ +
-&\rightarrow \frac{1}{3} ( 1 + 2x )( 1 + x + x^2 )^{ - \frac{2}{3} } +
-  +
-  +
-\end{split} +
-$$ +
-  +
-**c** +
-  +
-$$ +
-\begin{split}  +
- +
-( ( x^2 - 1 )^4 )' &\rightarrow 4 (x^2 - 1)^3 \cdot 2x \\ +
-&\rightarrow 8x(x^2 - 1)^3 +
-  +
-\end{split} +
-$$ +
-  +
-**d** +
-  +
-$$ +
-\begin{split}  +
-  +
-(\sqrt{ x^3 + 1 })' &\rightarrow \frac{1}{2} (x^3 + 1)^{ - \frac{1}{2} } \cdot 3x^2 \\ +
-&\rightarrow \frac{3}{2} x^2 (x^3 + 1)^{ - \frac{1}{2} } +
-  +
-\end{split} +
-$$ +
-  +
-**e** +
-  +
-$$ +
-\begin{split}  +
- +
-((x^2 + x)^{ \frac{3}{2} })' &\rightarrow \frac{3}{2} (x^2 + x)^{ \frac{1}{2} } \cdot (2x + 1) \\ +
-&\rightarrow \frac{3}{2} (2x + 1) (x^2 + x)^{ - \frac{1}{2} } +
-  +
-\end{split} +
-$$ +
-====== 20.13 ====== +
- +
-**b** +
- +
-$$ +
-\begin{split}  +
- +
-x \cos(2x) \\ +
-\cos (2x) &+ x - \sin(2x) \cdot 2 \\ +
-\cos (2x) &- 2x \sin(2x) +
- +
-\end{split} +
-$$ +
- +
-Behalve de productregel gebruik je hierbij ook de kettingregel! +
- +
- +
-====== 20.14 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
-\sqrt{x+1} \cdot \ln x& \rightarrow (x + 1)^{\frac{1}{2}} \cdot \ln x &+ \sqrt{x+1} \frac{1}{x} \\ +
-& \rightarrow \frac{1}{2} \cdot (x+1)^{-\frac{1}{2}} \cdot \ln x &+ \frac{\sqrt{x+1} \cdot 1}{x} \\ +
-& \rightarrow \frac{1}{2} \cdot \frac{1}{(x+1)^\frac{1}{2}} \cdot \ln x &+ \frac{\sqrt{x+1}}{x} \\ +
-& \rightarrow \frac{1}{2(x+1)^\frac{1}{2}} \cdot \frac{\ln x }{1} &+ \frac{\sqrt{x+1}}{x} \\ +
-& \rightarrow \frac{\ln x}{2 \sqrt{(x+1)}} &+ \frac{\sqrt{x+1}}{x} +
-\end{split} +
-$$ +
- +
-**c** +
- +
-$$ +
-\begin{split}  +
-x \ln \sqrt[3]{x} &  \rightarrow 1 x^0 \cdot \ln \sqrt[3]{x} &+ x \cdot \frac{1}{ \sqrt[3]{x} } \\ +
-& \rightarrow 1 \cdot \ln \sqrt[3]{x} &+ x \cdot \frac{1}{ x^\frac{1}{3} } \\ +
-& \rightarrow \ln \sqrt[3]{x} &+ x \cdot x^{-\frac{1}{3}} \\ +
-& \rightarrow \ln \sqrt[3]{x} &+ x^1 \cdot x^{-\frac{1}{3}} \\ +
-& \rightarrow \ln \sqrt[3]{x} &+ x^1 \cdot x^{-\frac{1}{3}}  +
-\end{split} +
-$$ +
- +
-Deze is nog niet correct, antwoord moet zijn: +
- +
-$ \ln \sqrt[3]{x} + \frac{1}{3} $+
jancraats/hoofdstuk_20.1349439200.txt.gz · Last modified: 2022/09/01 11:36 (external edit)