Sr Examen
Otras calculadoras
-
¿Cómo usar?
- La ecuación:
-
Ecuación 2*3^(x+1)-3^x=15
-
Ecuación -5x=10
-
Ecuación x^2+6*x+13=0
-
Ecuación 4x=2
- Expresar {x} en función de y en la ecuación:
- -17*x+9*y=-18
- -19*x-4*y=5
- -8*x-1*y=9
- 4*x+4*y=13
-
Expresiones idénticas
- (d*y)/(d*x)=(x^ dos (e^y+ uno))/e^y
- (d multiplicar por y) dividir por (d multiplicar por x) es igual a (x al cuadrado (e en el grado y más 1)) dividir por e en el grado y
- (d multiplicar por y) dividir por (d multiplicar por x) es igual a (x en el grado dos (e en el grado y más uno)) dividir por e en el grado y
- (d*y)/(d*x)=(x2(ey+1))/ey
- d*y/d*x=x2ey+1/ey
- (d*y)/(d*x)=(x²(e^y+1))/e^y
- (d*y)/(d*x)=(x en el grado 2(e en el grado y+1))/e en el grado y
- (dy)/(dx)=(x^2(e^y+1))/e^y
- (dy)/(dx)=(x2(ey+1))/ey
- dy/dx=x2ey+1/ey
- dy/dx=x^2e^y+1/e^y
- (d*y) dividir por (d*x)=(x^2(e^y+1)) dividir por e^y
-
Expresiones semejantes
- d*y/((d*x))=(3*x+y)/x
- (d*y)/(d*x)=(x^2(e^y-1))/e^y
(d*y)/(d*x)=(x^2(e^y+1))/e^y la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 / y \
d*y x *\E + 1/
--- = -----------
d*x y
E
$$\frac{d y}{d x} = \frac{x^{2} \left(e^{y} + 1\right)}{e^{y}}$$
Gráfica
Respuesta rápida
[src]
/ / / y \ / y \\\ / / / y \ / y \\\
| | | y*e | | y*e ||| | | | y*e | | y*e |||
___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||
/ / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||
/ 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|
x1 = / im |------| + re |------| *cos|-----------------------------| + I* / im |------| + re |------| *sin|-----------------------------|
6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /
\/ \1 + e / \1 + e / \/ \1 + e / \1 + e /
$$x_{1} = i \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)} + \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}$$
/ / / / y \ / y \\\ / / / y \ / y \\\\ / / / y \ / y \\\ / / / y \ / y \\\
| | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | y*e | | y*e ||| | | | y*e | | y*e |||
| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||
| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||
| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|
| / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| / im |------| + re |------| *cos|-----------------------------| \/ 3 * / im |------| + re |------| *sin|-----------------------------|
| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /
| \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e /
x2 = I*|- -------------------------------------------------------------------- - --------------------------------------------------------------------------| - -------------------------------------------------------------------- + --------------------------------------------------------------------------
\ 2 2 / 2 2
$$x_{2} = i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}$$
/ / / / y \ / y \\\ / / / y \ / y \\\\ / / / y \ / y \\\ / / / y \ / y \\\
| | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | y*e | | y*e ||| | | | y*e | | y*e |||
| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||
| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||
| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|
| / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| / im |------| + re |------| *cos|-----------------------------| \/ 3 * / im |------| + re |------| *sin|-----------------------------|
| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /
| \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e /
x3 = I*|- -------------------------------------------------------------------- + --------------------------------------------------------------------------| - -------------------------------------------------------------------- - --------------------------------------------------------------------------
\ 2 2 / 2 2
$$x_{3} = i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}$$
x3 = i*(-(re(y*exp(y)/(exp(y) + 1))^2 + im(y*exp(y)/(exp(y) + 1))^2)^(1/6)*sin(atan2(im(y*exp(y)/(exp(y) + 1), re(y*exp(y)/(exp(y) + 1)))/3)/2 + sqrt(3)*(re(y*exp(y)/(exp(y) + 1))^2 + im(y*exp(y)/(exp(y) + 1))^2)^(1/6)*cos(atan2(im(y*exp(y)/(exp(y) + 1)), re(y*exp(y)/(exp(y) + 1)))/3)/2) - sqrt(3)*(re(y*exp(y)/(exp(y) + 1))^2 + im(y*exp(y)/(exp(y) + 1))^2)^(1/6)*sin(atan2(im(y*exp(y)/(exp(y) + 1)), re(y*exp(y)/(exp(y) + 1)))/3)/2 - (re(y*exp(y)/(exp(y) + 1))^2 + im(y*exp(y)/(exp(y) + 1))^2)^(1/6)*cos(atan2(im(y*exp(y)/(exp(y) + 1)), re(y*exp(y)/(exp(y) + 1)))/3)/2)
Suma y producto de raíces
[src]
suma
/ / / / y \ / y \\\ / / / y \ / y \\\\ / / / y \ / y \\\ / / / y \ / y \\\ / / / / y \ / y \\\ / / / y \ / y \\\\ / / / y \ / y \\\ / / / y \ / y \\\
| | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | y*e | | y*e ||| | | | y*e | | y*e ||| | | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | y*e | | y*e ||| | | | y*e | | y*e |||
| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------||| | ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||
/ / / y \ / y \\\ / / / y \ / y \\\ | / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y||| | / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||
| | | y*e | | y*e ||| | | | y*e | | y*e ||| | / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| | / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|
___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------||| | / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| / im |------| + re |------| *cos|-----------------------------| \/ 3 * / im |------| + re |------| *sin|-----------------------------| | / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| / im |------| + re |------| *cos|-----------------------------| \/ 3 * / im |------| + re |------| *sin|-----------------------------|
/ / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y||| | 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 / | 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /
/ 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e /
/ im |------| + re |------| *cos|-----------------------------| + I* / im |------| + re |------| *sin|-----------------------------| + I*|- -------------------------------------------------------------------- - --------------------------------------------------------------------------| - -------------------------------------------------------------------- + -------------------------------------------------------------------------- + I*|- -------------------------------------------------------------------- + --------------------------------------------------------------------------| - -------------------------------------------------------------------- - --------------------------------------------------------------------------
6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 / \ 2 2 / 2 2 \ 2 2 / 2 2
\/ \1 + e / \1 + e / \/ \1 + e / \1 + e /
$$\left(\left(i \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)} + \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}\right) + \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right)\right) + \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right)$$
=
/ / / / y \ / y \\\ / / / y \ / y \\\\ / / / / y \ / y \\\ / / / y \ / y \\\\
| | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | | y*e | | y*e ||| | | | y*e | | y*e ||||
| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| | ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------||||
| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| | / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / / y \ / y \\\
| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| | / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| | | | y*e | | y*e |||
| / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| | / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| ___________________________ |atan2|im|------|, re|------|||
| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| | 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| / / y \ / y \ | | | y| | y|||
| \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|
I*|- -------------------------------------------------------------------- + --------------------------------------------------------------------------| + I*|- -------------------------------------------------------------------- - --------------------------------------------------------------------------| + I* / im |------| + re |------| *sin|-----------------------------|
\ 2 2 / \ 2 2 / 6 / | y| | y| \ 3 /
\/ \1 + e / \1 + e /
$$i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) + i \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}$$
producto
/ / / / / y \ / y \\\ / / / y \ / y \\\\ / / / y \ / y \\\ / / / y \ / y \\\\ / / / / / y \ / y \\\ / / / y \ / y \\\\ / / / y \ / y \\\ / / / y \ / y \\\\
| | | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | | y*e | | y*e ||| | | | y*e | | y*e ||||
| | ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| | | ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------||||
/ / / / y \ / y \\\ / / / y \ / y \\\\ | | / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| | | / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y||||
| | | | y*e | | y*e ||| | | | y*e | | y*e |||| | | / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| | | / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| ___ / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //||
| ___________________________ |atan2|im|------|, re|------||| ___________________________ |atan2|im|------|, re|------|||| | | / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| / im |------| + re |------| *cos|-----------------------------| \/ 3 * / im |------| + re |------| *sin|-----------------------------|| | | / im |------| + re |------| *sin|-----------------------------| \/ 3 * / im |------| + re |------| *cos|-----------------------------|| / im |------| + re |------| *cos|-----------------------------| \/ 3 * / im |------| + re |------| *sin|-----------------------------||
| / / y \ / y \ | | | y| | y||| / / y \ / y \ | | | y| | y|||| | | 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| | | 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| 6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /|
| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //| / 2| y*e | 2| y*e | | \ \1 + e / \1 + e //|| | | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | | | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / | \/ \1 + e / \1 + e / \/ \1 + e / \1 + e / |
| / im |------| + re |------| *cos|-----------------------------| + I* / im |------| + re |------| *sin|-----------------------------||*|I*|- -------------------------------------------------------------------- - --------------------------------------------------------------------------| - -------------------------------------------------------------------- + --------------------------------------------------------------------------|*|I*|- -------------------------------------------------------------------- + --------------------------------------------------------------------------| - -------------------------------------------------------------------- - --------------------------------------------------------------------------|
|6 / | y| | y| \ 3 / 6 / | y| | y| \ 3 /| \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
\\/ \1 + e / \1 + e / \/ \1 + e / \1 + e / /
$$\left(i \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)} + \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}\right) \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{2}\right)$$
=
/ / / y \ / / / y \ / y \\\\ \
| | | y*e | | | | y*e | | y*e |||| |
| | im|------| |atan2|im|------|, re|------|||| |
| | | y| | | | y| | y|||| |
| | \1 + e / | \ \1 + e / \1 + e //|| |
| / / / y \ / y \\\ 3*I*|--------------------------------- + sin|-----------------------------|| |
| | | | y*e | | y*e ||| | ___________________________ \ 3 /| |
| / / / y \ / y \\\ |atan2|im|------|, re|------||| / / / y \ / y \\\ | / / y \ / y \ | / y \ |
| | | | y*e | | y*e ||| | | | y| | y||| | | | y*e | | y*e ||| | / 2| y*e | 2| y*e | | | y*e | |
___________________________ | |atan2|im|------|, re|------||| | \ \1 + e / \1 + e //| |atan2|im|------|, re|------||| | / im |------| + re |------| | 3*re|------| |
/ / y \ / y \ | | | | y| | y||| 3*cos|-----------------------------| | | | y| | y||| | / | y| | y| | | y| |
/ 2| y*e | 2| y*e | | 3| \ \1 + e / \1 + e //| \ 3 / 3| \ \1 + e / \1 + e //| \\/ \1 + e / \1 + e / / \1 + e / |
/ im |------| + re |------| *|cos |-----------------------------| - ------------------------------------ - I*sin |-----------------------------| + ---------------------------------------------------------------------------- + -----------------------------------|
/ | y| | y| | \ 3 / 4 \ 3 / 4 ___________________________|
\/ \1 + e / \1 + e / | / / y \ / y \ |
| / 2| y*e | 2| y*e | |
| 4* / im |------| + re |------| |
| / | y| | y| |
\ \/ \1 + e / \1 + e / /
$$\sqrt{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)} + \frac{\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}}{\sqrt{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}}}\right)}{4} - i \sin^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)},\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)} \right)}}{3} \right)}}{4} + \frac{3 \operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}}{4 \sqrt{\left(\operatorname{re}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{y e^{y}}{e^{y} + 1}\right)}\right)^{2}}}\right)$$
sqrt(im(y*exp(y)/(1 + exp(y)))^2 + re(y*exp(y)/(1 + exp(y)))^2)*(cos(atan2(im(y*exp(y)/(1 + exp(y))), re(y*exp(y)/(1 + exp(y))))/3)^3 - 3*cos(atan2(im(y*exp(y)/(1 + exp(y))), re(y*exp(y)/(1 + exp(y))))/3)/4 - i*sin(atan2(im(y*exp(y)/(1 + exp(y))), re(y*exp(y)/(1 + exp(y))))/3)^3 + 3*i*(im(y*exp(y)/(1 + exp(y)))/sqrt(im(y*exp(y)/(1 + exp(y)))^2 + re(y*exp(y)/(1 + exp(y)))^2) + sin(atan2(im(y*exp(y)/(1 + exp(y))), re(y*exp(y)/(1 + exp(y))))/3))/4 + 3*re(y*exp(y)/(1 + exp(y)))/(4*sqrt(im(y*exp(y)/(1 + exp(y)))^2 + re(y*exp(y)/(1 + exp(y)))^2)))