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jancraats:hoofdstuk_20 [2012/10/01 18:19] – [20.12] jerryjancraats:hoofdstuk_20 [2022/09/01 11:44] (current) – external edit 127.0.0.1
Line 1: Line 1:
-  * Let op de **\rightarrow** (latex-opmaak) bij differentieren, gebruik geen gelijk-aan teken. Functies zijn niet gelijk aan elkaar, ze worden getransformeerd.+<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>
  
-====== 20.1 ====== +  * [[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]] 
-**a** +  * [[jancraats/hoofdstuk_20/pagina_180 | Pagina 180 - Opgaves van 20.19 tot 20.31]]
- +
-$$ +
-\begin{split}  +
-2x-3 &\rightarrow 2x^{1} \\ +
-&\rightarrow 2 +
-\end{split} +
-$$ +
- +
-**b** +
- +
-$$ +
-\text{Constant value not able to differentatie goes to 0} +
-$$ +
- +
-**c** +
- +
-$$ +
-\begin{split}  +
-4x^{2}+1 &\rightarrow 4x^{2-1} \\ +
-&\rightarrow 2\cdot4x^1 \\ +
-&\rightarrow 8x +
-\end{split} +
-$$ +
- +
-**d** +
- +
-$$ +
-\begin{split}  +
-10x^7 &\rightarrow 10x^{7-1} \\ +
-&\rightarrow 7 \cdot 10x^6 \\ +
-&\rightarrow 70x^6 +
-\end{split} +
-$$ +
- +
-**e** +
- +
-$$ +
-\begin{split}  +
-4x + x^3 &\rightarrow 4 + 3x^2 +
-\end{split} +
-$$ +
- +
-====== 20.2 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
-x^3 - 3 &\rightarrow 3x^{3-1} \\ +
-&\rightarrow 3x^2 +
-\end{split} +
-$$ +
- +
-**b** +
- +
-$$ +
-\begin{split}  +
-x^2 - 2x + 1 &\rightarrow 2x^{2-1}-2 \\ +
-&\rightarrow 2x - 2 +
-\end{split} +
-$$ +
- +
-**c** +
- +
-$$ +
-\begin{split}  +
-x^4 - 3x^3 + 2 &\rightarrow x^4 - 3x^3 \\ +
-&\rightarrow 4x^3 - 3 \cdot 3x^2 \\ +
-&\rightarrow 4x^3 - 9x^2 +
-\end{split} +
-$$ +
- +
-**d** +
- +
-$$ +
-\begin{split}  +
-8x^8 &\rightarrow 8 \cdot 8x^{8-1} \\ +
-&\rightarrow 64x^7 +
-\end{split} +
-$$ +
- +
-**e** +
- +
-$$ +
-\begin{split}  +
-x^6 - 6x^4 &\rightarrow 6x^{6-1} - 4 \cdot 6x^{4-1} \\ +
-&\rightarrow 6x^5 - 24^3 +
-\end{split} +
-$$ +
- +
-====== 20.3 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
-& 4x^4 -3x^2 + 2 \\ +
-& 16x^3 -6x +
-\end{split} +
-$$ +
- +
-**e** +
- +
-$$ +
-\begin{split}  +
-x^2 -5x^3 +1 \\ +
-2x -15x^2 +1 +
-\end{split} +
-$$ +
-====== 20.4 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
-\sqrt{x} &\rightarrow x^{\frac{1}{2}} \\ +
-&\rightarrow \frac{1}{2}x^{-\frac{1}{2}} +
-\end{split} +
-$$ +
- +
-**b** +
- +
-$$ +
-\begin{split}  +
-x\sqrt{x} &\rightarrow x^1 \cdot x^{\frac{1}{2}} \\ +
-&\rightarrow x^{\frac{2}{2}} \cdot x^{\frac{1}{2}} \\ +
-&\rightarrow x^{\frac{3}{2}} \\ +
-&\rightarrow \frac{3}{2} \cdot x^{\frac{3}{2}-1} \\ +
-&\rightarrow  \frac{3}{2} x^{\frac{1}{2}} +
-\end{split} +
-$$ +
- +
-**c** +
- +
-$$ +
-\begin{split} +
- +
-\sqrt{x^3} &\rightarrow (x^3)^{\frac{1}{2}} \\ +
-&\rightarrow x^{\frac{3}{1} \cdot \frac{1}{2}} \\ +
-&\rightarrow \frac{3}{2} x^{ \frac{3}{2} - 1} \\ +
-&\rightarrow \frac{3}{2} x^{ \frac{1}{2} } +
- +
-\end{split} +
-$$ +
- +
-**d** +
- +
-$$ +
-\begin{split} +
- +
-x^2 \sqrt{x} &\rightarrow x^2 \cdot x^{ \frac{1}{2} } \\ +
-&\rightarrow  +
- +
-\end{split} +
-$$ +
- +
-**e** +
- +
-====== 20.5 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
- +
-\sqrt[3]{x} \\ +
-x^{\frac{1}{3}} \\ +
-\frac{1}{2} x ^ {-\frac{2}{3}} +
- +
-\end{split} +
-$$ +
- +
-**e** +
- +
-$$ +
-\begin{split}  +
- +
-x^2 \sqrt[5]{x^2} \\ +
-x^2 x ^ {\frac{2}{5}} \\ +
-x^ {\frac{12}{5}} \\ +
-\frac{12}{5} x ^ {\frac{7}{5}} +
- +
-\end{split} +
-$$ +
- +
-====== 20.6 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
- +
-\sqrt[7]{x^2} \\ +
-x ^ {\frac{2}{7}} \\ +
-\frac{2}{7} \\ +
-\frac{2}{7} x ^ {-\frac{5}{7}} +
- +
-\end{split} +
-$$ +
- +
-**b** +
- +
-$$ +
-\begin{split}  +
- +
-\sqrt{3x^3} \\ +
-3^{\frac{1}{2}} \cdot (x3)^{\frac{1}{2}} \\ +
-3^{\frac{1}{2}} \cdot x^{\frac{3}{2}} \\ +
-\frac{3}{2} \sqrt{3} x ^ {\frac{1}{2}} +
- +
-\end{split} +
-$$ +
- +
-====== 20.9 ====== +
- +
-**a** +
- +
-$$ +
-\begin{split}  +
- +
-\frac{1}{x} &\rightarrow x^{-1} \\ +
-&\rightarrow -1x^{-1-1} \\ +
-&\rightarrow -1x^{-2} \\ +
-&\rightarrow -\frac{2}{x} +
- +
-\end{split} +
-$$ +
- +
-**b** +
- +
-$$ +
-\begin{split}  +
- +
-\frac{3}{2x} &\rightarrow \frac{3}{2} \cdot \frac{1}{x} \\ +
-&\rightarrow \frac{3}{2} \cdot x^{-1} \\ +
-&\rightarrow \frac{3}{2} \cdot -1x^{-1-1} \\ +
-&\rightarrow -\frac{3}{2} x^{-2} +
- +
- +
-\end{split} +
-$$ +
- +
-**c** +
- +
-$$ +
-\begin{split}  +
- +
-\frac{5}{x^5} &\rightarrow \frac{5}{1} \cdot \frac{1}{x^5} \\ +
-&\rightarrow 5 \cdot x^{-5} \\ +
-&\rightarrow 5 \cdot 5x^{-5-1} \\ +
-&\rightarrow -25x^{-6} +
- +
-\end{split} +
-$$ +
- +
-**d** +
- +
-$$ +
-\begin{split}  +
- +
-\frac{ \sqrt{x} }{ x } &\rightarrow \frac{ \sqrt{x} \cdot 1 }{ 1 \cdot x } \\ +
-&\rightarrow \sqrt{x} \cdot \frac{1}{x} \\ +
-&\rightarrow \sqrt{x} \cdot x^{-1} \\ +
-&\rightarrow x^{ \frac{1}{2} } \cdot x^{-1} \\ +
-&\rightarrow x^{ (\frac{1}{2} - 1) } \\ +
-&\rightarrow x^{ -\frac{1}{2} } \\ +
-&\rightarrow - \frac{1}{2} x^{ -\frac{1}{2}-\frac{2}{2} } \\ +
-&\rightarrow - \frac{1}{2} x^{ -\frac{3}{2} } \\ +
-\end{split} +
-$$ +
- +
- +
-**e** +
- +
-$$ +
-\begin{split}  +
- +
-\frac{1}{x\sqrt[3]{x}} \\ +
-1 \cdot \frac{1}{ (x \sqrt[3]{x}) ^ 1 } \\ +
-1 \cdot (x \sqrt[3]{x}) ^ {-1} \\ +
-1 \cdot x^{-1} \cdot \sqrt[3]{x} ^ {-1} \\ +
-x^{-1} \cdot (x^{\frac{1}{3}}) ^ {-1} \\ +
-x^{-1} \cdot x^{-\frac{1}{3}} \\ +
-x^{-\frac{4}{3}} \\ +
--\frac{4}{3} x^{-\frac{7}{3}} +
- +
-\end{split} +
-$$ +
- +
-====== 20.10 ====== +
- +
-**e** +
- +
-$$ +
-\begin{split}  +
- +
-(x - x^4) ^ {-2} \\ +
--2 (x - x^4) ^ {-3} \cdot 1 - 4 x^3 \\ +
--2(x - x^4) ^ {-3} +
- +
-\end{split} +
-$$ +
- +
-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_20.1349115591.txt.gz · Last modified: 2022/09/01 11:36 (external edit)