produkt-und-kettenregel

$$||"id":1498371211072||f(x)=x^n{f}^{\prime }(x)=n\ast x^{n-1}f(x)=\sin (x)f^{\prime }(x)=\cos (x)$$

Produktregel

$(f\ast g{)}^{\prime }=f^{\prime }\ast g+f\ast g^{\prime }$

Quotientenregel

$(\frac{f}{g}{)}^{\prime }=\frac{f^{\prime }\ast g-f\ast g^{\prime }}{g^2}$g(x)!=0

Kettenregel:

$h(x)=f(g(x))$ $\Rightarrow h^{\prime }(x)=f^{\prime }(g(x))\ast g^{\prime }(x)$

Produktregel bsp

$$||"id":1635603873474||f_1(x)=x\ast \sin (x)f_1^{\prime }(x)=(x\ast \sin x{)}^{\prime }\ne 1\ast \cos x{f}_1^{\prime }(x)=(x{)}^{\prime }\ast \sin x+x\ast (\sin x{)}^{\prime }{f}_1^{\prime }(x)=1\ast \sin x+1\ast \cos x{f}_1^{\prime }(x)=\sin x+c\ast \cos (x)$$

Kettenregel bsp

$$||"id":538972515398||f_2(x)=\sin (\frac{1}{x})f_2(x)=\sin (x^{-1})f_2(x)=\sin (x^{-1})f_2^{\prime }(x)=\cos (x^{-1})\ast (-1)\ast x^{-2}{f}_2^{\prime }(x)=-\cos (x^{-1})\ast x^{-2}$$