Was ist die 101. Ableitung von f(x)=cos(x)?
\(\ f(x)\ =cos(x),\ f'(x)\ =-sin(x),\ f''(x)\ = -cos(x), f^{3'}(x)=sin(x)\)
\(f^{4'}(x)=cos(x),\ f^{5'}(x)=-sin(x),\ f^{6'}(x)=-cos(x), f^{7'}(x)=sin(x)\)
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\(\ f^{96'}(x)=\) .. .. \(f^{99'}(x)=sin(x)\)
\(f^{100'}(x)=cos(x),\ f^{101'}(x)=-sin(x),\ f^{102'}(x)=-cos(x), f^{103'}(x)=sin(x)\)
\(n\in \mathbb{N}\\ f(x)=cos(x)\\ f^{n'}=cos(x)\ |\frac{n}{4}\in \mathbb{N}\\ f^{n'}=-sin(x)\ |\frac{n+3}{4}\in \mathbb{N}\\ f^{n'}=-cos(x)\ |\frac{n+2}{4}\in \mathbb{N}\\ f^{n'}=sin(x)\ |\frac{n+1}{4}\in \mathbb{N}\\\)
\(f^{101'}=-sin(x)\ |(\frac{101+3}{4}=26)\in \mathbb{N}\)
Die 101. Ableitung von f(x) = cos(x)
ist
\(f^{101'}(x)=-sin(x)\)
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