# Benutzer:Exxu/InArbeit06

Zur Navigation springen Zur Suche springen
(a)

${\displaystyle g(t)=9-t^{2};\ g'(t)=-2t;\ g''(t)=-2;\ g''(-1)=-2;}$

(b)

${\displaystyle f(x)=(5-2x)^{\frac {1}{2}};}$

${\displaystyle u(x)=x^{\frac {1}{2}};\ v(x)=(5-2x);\ v'(x)=-2;}$

${\displaystyle v(-2)=9;v'(-2)=-2;v''(-2)=v'''(-2)=0;\ }$

${\displaystyle u'(x)={\frac {1}{2}}x^{\frac {-1}{2}};\ u''(x)={\frac {-1}{4}}x^{\frac {-3}{2}};u'''(x)={\frac {3}{8}}x^{\frac {-5}{2}};}$

${\displaystyle u'''(9)={\frac {1}{8*3^{4}}};\ }$

${\displaystyle f(x)=u(v(x));\ f'(x)=u'(v(x))v'(x);}$

${\displaystyle f''(x)=u''(v(x))v'^{2}(x)+u'(v(x))v''(x);\ }$

${\displaystyle f'''(x)=u'''(v(x))v'^{3}(x)+u''(v(x))2v'(x)v''(x)+u''(v(x))v'(x)v''(x)+u'(v(x))v'''(x);\ }$

${\displaystyle f'''(-2)=u'''(9)(-8)={\frac {-1}{3^{4}}}={\frac {-1}{81}};\ }$

(c)

${\displaystyle f(t)=cos(\pi (1-3t)^{\frac {1}{2}});\ }$

${\displaystyle u(x)=cos(x);v(x)=\pi (1-3x)^{\frac {1}{2}};\ }$

${\displaystyle u'(x)=-sin(x);v'(x)={\frac {-3}{2}}\pi (1-3x)^{\frac {-1}{2}};\ }$

${\displaystyle u''(x)=-cos(x);\ }$

${\displaystyle v(0)=\pi ;\ v'(0)=\pi {\frac {-3}{2}};\ }$

${\displaystyle u'(\pi )=0;\ u''(\pi )=1;}$

${\displaystyle f(t)=u(v(t));\ f'(t)=u'(v(t))v'(t);}$

${\displaystyle g(x)=u'(v(x));\ h(x)=v'(x);}$

${\displaystyle g'(x)=u''(v(x))v'(x);\ h'(x)=v''(x);}$

${\displaystyle f''(t)=(g(t)h(t))'=g'(t)h(t)+g(t)h'(t);\ }$

${\displaystyle f''(0)=g'(0)h(0)+g(0)h'(0)=u''(v(0))(v'(0))^{2}+u'(v(0))v''(0);\ }$

${\displaystyle f''(0)=1(\pi {\frac {-3}{2}})^{2}+0={\frac {9}{4}}\pi ^{2}\ }$

a)

${\displaystyle f(x)=x^{2}\exp {x};\ f'(x)=\exp {x}(2x+x^{2});}$

b)

${\displaystyle f(x)={\frac {1}{2}}(e^{(x)}+e^{(-x)});\ f'(x)={\frac {1}{2}}(e^{(x)}-e^{(-x)});}$

c)

${\displaystyle g(t)={\frac {5}{1+3e^{(-0.1t)}}};\ g'(t)={\frac {1.5e^{(-0.1t)}}{(1+3e^{(-0.1t)})^{2}}};}$

a)

${\displaystyle f(x)=x^{2}+4x-10;\ }$

${\displaystyle f'(x)=\lim _{h}{\frac {(x+h)^{2}+4(x+h)-10-(x^{2}+4x-10)}{h}};\ }$

${\displaystyle f'(x)=\lim _{h}{(2x+h+4)}=2x+4;\ }$

b)

${\displaystyle f(x)={\frac {1}{4x+1}};\ }$

${\displaystyle f'(x)=\lim _{h}{\frac {{\frac {1}{4(x+h)+1}}-{\frac {1}{4x+1}}}{h}};\ }$

${\displaystyle f'(x)=\lim _{h}{{\frac {4x+1-(4(x+h)+1)}{(4(x+h)+1)(4x+1)}}{\frac {1}{h}}};\ }$

${\displaystyle f'(x)=\lim _{h}{\frac {4}{(4(x+h)+1)(4x+1)}}={\frac {4}{(4x+1)^{2}}};\ }$

c)

${\displaystyle f(x)=sin(x);\ }$

${\displaystyle f'(x)=\lim _{h}{\frac {sin(x+h)-sin(x)}{h}}=\lim _{h}{\frac {sin(x)cos(h)+cos(x)sin(h)-sin(x)}{h}};\ }$

${\displaystyle f'(x)=\lim _{h}(sin(x){\frac {cos(h)-1}{h}}+cos(x){\frac {sin(h)}{h}})=sin(x)*0+cos(x)*1;\ }$

${\displaystyle f(x)=x^{5}sin(x)=u(x)v(x);\ u(x)=x^{5};v(x)=sin(x);}$

${\displaystyle u'(x)=5x^{4};\ u''(x)=20x^{3};...}$

${\displaystyle v'=cos(x);\ v''=-sin(x);...\ }$

${\displaystyle f'(x)=u'v+uv';f''=u''v+2u'v'+uv'';\ }$

${\displaystyle f'''(x)=u'''v+3u''v'+3u'v''+uv''';\ }$

${\displaystyle f''''=u''''v+4u'''v'+6u''v''+4u'v'''+uv'''';\ }$

Koeffizient vor ${\displaystyle x^{3}sin(x):\ 6u''v'':\ -120\ ;}$

${\displaystyle y(x)={\frac {x^{2}+20x+100}{x}};\ }$

${\displaystyle y'={\frac {(2x+20)x-(x^{2}+20x+100)1}{x^{2}}}={\frac {x^{2}-100}{x^{2}}};\ }$

${\displaystyle y'(10)=0;\ y(10)=40;\ }$ Gleichung der Tangente an der Stelle ${\displaystyle x=10:f(x)=40;\ }$