ableitungsregeln-übungen

nur zahlen fallen simply weg

potenzregel

$$||"id":899360649066||f(x)=x^3{f}^{\prime }(x)=3\ast x^{2}f(x)=x^{1000}{f}^{\prime }(x)=1000\ast x^{999}f(x)=x^{-7}{f}^{\prime }(x)=-7\ast x^8$$

Faktorregel

$$||"id":477983730201||f(x)=3\ast x^5{f}^{\prime }(x)=3\ast 5\ast x^4=15\ast x^{4}f(t)=\sqrt{2}\ast 4\ast t^{3}f(x)=\frac{x^3}{5}=\frac{1}{5}\ast x^3{f}^{\prime }(x)=\frac{1}{5}\ast 3x^2$$

summenregel

$$||"id":666571568885||f(x)=x^2-5x^3+7f^{\prime }(x)=2x-5\ast 3\ast x^2+0=2x-15x^2$$

produkt regel

$f^{\prime }(x)=u^{\prime }\ast v+u\ast v^{\prime }$

$$||"id":463126828029||f(x)=x^3\ast x^{5}u=x^{3}v=x^5{u}^{\prime }=3x^{2}v=5x^4{f}^{\prime }(x)=3x^2\ast x^5+x^3+5x^4=3x^7+5x^7=8x^7$$

Quotientenregel

$f(x)=\frac{u^{\prime }\ast v-u\ast v^{\prime }}{v^2}$

$$||"id":139569537811||f(x)=\frac{2x+3}{x^5}u=2x+3=2\ast 1+0=2v=x^5=5\ast x^4{f}^{\prime }(x)=\frac{2\ast x^5-(2x+3)\ast 5x^4}{(x^5{)}^2}$$

Kettenregel

$$||"id":1538055582617||f(x)=(x^4+5{)}^7$$