Hello dear community,
how do I manually derive the functions f(x)= \(\frac{A}{\sqrt{x²+y²+z²}}\) and w(f)=sin(2\(\varpi \)f)-cos(4\(\varpi \)f) ?
Greetings, Simon
+14983 How do I manually derive the functions \(f(x)=\frac{A}{\sqrt{x^2+y^2+z^2}}\) and \(w(f)=sin(2\overline \omega f)-cos(4\overline\omega f)\) ?
Wie leite ich die Funktionen \(f(x)=\frac{A}{\sqrt{x^2+y^2+z^2}}\) und \(w(f)=sin(2\overline \omega f)-cos(4\overline\omega f)\) manuell ab?
Hi Simon!
I presuppose (ich setze voraus): \(f(x) \neq y\)
\(\color{BrickRed}f(x)=\frac{A}{\sqrt{x^2+y^2+z^2}}\\ f(x)=A\cdot (x^2+y^2+z^2)^{-\frac{1}{2}}\\ \frac{f(x)}{dx}=A\cdot (-\frac{1}{2})\cdot (x^2+y^2+z^2)^{-\frac{3}{2}}\cdot 2x\)
\(\frac{f(x)}{dx}=\frac{-A\cdot x}{\sqrt{(x^2+y^2+z^2)^3}}\)
\(w(f)=sin(2\overline\omega f)-cos(4\overline\omega f)\)
\(\frac{w(f)}{df}=2\overline\omega\cdot cos(2\overline \omega f)-(-sin(4\overline\omega f)\cdot 4\overline\omega)\)
\(\frac{w(f)}{df}=2\overline\omega\cdot cos(2\overline\omega f)+4\overline\omega\cdot sin(4\overline\omega f)\)
\(\frac{w(f)}{df}=2\overline\omega\cdot [cos(2\overline\omega f)+2\cdot sin(4\overline\omega f)]\)
\(\overline \omega\) = 2,62205755... ist die Lemniskatische Konstante von Gauss.
!
