Berechnen Sie die Ableitung der folgenden Funktionen. Berechnen Sie auch den ungefähren Wert der Funktion und ihrer Ableitung an der Stelle 2,4. A) b ( v ) = 7 sin v − ln v + 3 5 v 3 + 3 {\displaystyle \ b(v)=7\ \sin v-\ln v+{\frac {3}{5\ v^{3}}}+3} B) v ( t ) = 5 t 3 4 − cos t + t − t − 3 {\displaystyle \ v(t)={\frac {5}{\sqrt[{4}]{t^{3}}}}-\cos t+t-t^{-3}} C) t ( b ) = 5 e b − 27 b 7 3 + tan b {\displaystyle \ t(b)=5\ e^{b}-{\sqrt[{3}]{\frac {27}{b^{7}}}}+\tan b\quad }
A) 3 5 v 3 = 3 5 v − 3 ⇒ {\displaystyle \ \ {\frac {3}{5\ v^{3}}}={\frac {3}{5}}\ v^{-3}\ \Rightarrow \ } b ( v ) = 7 sin v − ln v + 3 5 v − 3 + 3 ⇒ {\displaystyle \qquad b(v)=7\ \sin v-\ln v+{\frac {3}{5}}\ v^{-3}+3\ \Rightarrow \ } b ′ ( v ) = 7 cos v − 1 v − 9 5 v − 4 {\displaystyle \qquad b'(v)=7\cos v-{\frac {1}{v}}-{\frac {9}{5}}\ v^{-4}} b ( 2 , 4 ) ≈ 6 , 90 b ′ ( 2 , 4 ) ≈ − 2 , 45 {\displaystyle \qquad b(2{,}4)\approx 6{,}90\qquad b'(2{,}4)\approx -2{,}45}
B) 5 t 3 4 = 5 t − 3 4 ⇒ {\displaystyle \ \ {\frac {5}{\sqrt[{4}]{t^{3}}}}={5}\ t^{-{\frac {3}{4}}}\ \Rightarrow \ } v ( t ) = 5 t − 3 4 − cos t + t − t − 3 ⇒ {\displaystyle \qquad v(t)={5}\ t^{-{\frac {3}{4}}}-\cos t+t-t^{-3}\ \Rightarrow \ } v ′ ( t ) = − 15 4 t − 7 4 + sin t + 1 + 3 t − 4 {\displaystyle \qquad v'(t)={-{\frac {15}{4}}}\ t^{-{\frac {7}{4}}}+\sin t+1+3\ t^{-4}\ } v ( 2 , 4 ) ≈ 5 , 66 v ′ ( 2 , 4 ) ≈ − 1 , 15 {\displaystyle \qquad v(2{,}4)\approx 5{,}66\qquad v'(2{,}4)\approx -1{,}15}
C) 27 b 7 3 = 3 t − 7 3 ⇒ {\displaystyle \ \ {\sqrt[{3}]{\frac {27}{b^{7}}}}={3}\ t^{-{\frac {7}{3}}}\ \Rightarrow \ } t ( b ) = 5 e b − 3 b − 7 3 + tan b {\displaystyle \qquad t(b)=5\ e^{b}-{3}\ b^{-{\frac {7}{3}}}+\tan b\quad } t ′ ( b ) = 5 e b + 7 b − 10 3 + 1 cos 2 b {\displaystyle \qquad t'(b)=5\ e^{b}+{7}\ b^{-{\frac {10}{3}}}+{\frac {1}{\cos ^{2}b}}\quad }
t ( 2 , 4 ) ≈ 53 , 8 t ′ ( 2 , 4 ) ≈ 57 , 3 {\displaystyle \qquad t(2{,}4)\approx 53{,}8\qquad t'(2{,}4)\approx 57{,}3}
![{\displaystyle \ \ {\frac {5}{\sqrt[{4}]{t^{3}}}}={5}\ t^{-{\frac {3}{4}}}\ \Rightarrow \ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac277e6aa4a3a6ff3417e491b3dd9a5dd36245c)
![{\displaystyle \ \ {\sqrt[{3}]{\frac {27}{b^{7}}}}={3}\ t^{-{\frac {7}{3}}}\ \Rightarrow \ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a96d47d27c26ea7960c0f02a0baf901f94102e9)
![{\displaystyle \ v(t)={\frac {5}{\sqrt[{4}]{t^{3}}}}-\cos t+t-t^{-3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/200551b46994e1983804d38546981a8f3e987a65)
![{\displaystyle \ t(b)=5\ e^{b}-{\sqrt[{3}]{\frac {27}{b^{7}}}}+\tan b\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1ba99fe562fa818ef934295eb4049419a6c1ae)