$f\left(x\right)=\ln \left(\sqrt{x}\right)f\backslash left(x\backslash right)=\backslash ln\backslash left(\backslash sqrt\; x\backslash right)$

$g\left(x\right)=\ln \left(x\right)h\left(x\right)=\sqrt{x}\Rightarrow f\left(x\right)=g\left(h\left(x\right)\right)g\backslash left(x\backslash right)=\backslash ln\backslash left(x\backslash right)\backslash \backslash h\backslash left(x\backslash right)=\backslash sqrt\; x\backslash \backslash \backslash Rightarrow\; f\backslash left(x\backslash right)=g\backslash left(h\backslash left(x\backslash right)\backslash right)$

$g^{\mathrm{\prime }}\left(x\right)=\frac{1}{x}{h}^{\mathrm{\prime }}\left(x\right)=\frac{1}{2\sqrt{x}}g\text{'}\backslash left(x\backslash right)=\backslash frac\{1\}\{x\}\backslash \backslash h\text{'}\backslash left(x\backslash right)=\backslash frac\{1\}\{2\backslash sqrt\; x\}$

Setze nun alles in die Formel der Kettenregel ein

Zerlege $ff$ so, dass du die Kettenregel anwenden kannst.

$f(x)=e^{\sin (x^2)}f(x)=e^\{\backslash sin(x^2)\}$

Um die Ableitung von $ff$ anzugeben, muss man die Ableitungen von $gg$ und $hh$ bestimmen.

$gg$ kann direkt abgeleitet werden, um $hh$ abzuleiten, muss die Kettenregel erneut verwendet werden. Zerlege dazu $hh$.

$g(x)=e^{x}h(x)=\sin (x^2)\Rightarrow f(x)=g(h(x))g(x)=e^x\backslash \backslash h(x)=\backslash sin(x^2)\backslash \backslash \backslash Rightarrow\; f(x)=g(h(x))$

Leite $uu$ und $vv$ ab.

$g^{\mathrm{\prime }}(x)=e^{x}g\text{'}(x)=e^x$

$u(x)=\sin (x)v(x)=x^2\Rightarrow h(x)=u(v(x))u(x)=\backslash sin(x)\backslash \backslash v(x)=x^2\backslash \backslash \backslash Rightarrow\; h(x)=u(v(x))$

$u^{\mathrm{\prime }}(x)=\cos (x)v^{\mathrm{\prime }}(x)=2xu\text{'}(x)=\backslash cos(x)\backslash \backslash v\text{'}(x)=2x$