Sr Examen

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Derivada de la función/ x^n-x/ |x^n-x^(3n)|

Derivada de |x^n-x^(3n)|

Función f() - derivada -er orden en el punto
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
| n    3*n|
|x  - x   |
∣−x3n+xn∣\left|{- x^{3 n} + x^{n}}\right| 
Abs(x^n - x^(3*n))
Primera derivada [src] 
/                      /  /   n\       /   3*n\\                         /  /   n\       /   3*n\\\                
|/    / 3*n\     / n\\ |im\n*x /   3*im\n*x   /|   /    / 3*n\     / n\\ |re\n*x /   3*re\n*x   /||     / 3*n    n\
|\- im\x   / + im\x //*|-------- - ------------| + \- re\x   / + re\x //*|-------- - ------------||*sign\x    - x /
\                      \   x            x      /                         \   x            x      //                
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                                                      3*n    n                                                     
                                                     x    - x
((re⁡(nxn)x−3re⁡(nx3n)x)(re⁡(xn)−re⁡(x3n))+(im⁡(nxn)x−3im⁡(nx3n)x)(im⁡(xn)−im⁡(x3n)))sign⁡(x3n−xn)x3n−xn\frac{\left(\left(\frac{\operatorname{re}{\left(n x^{n}\right)}}{x} - \frac{3 \operatorname{re}{\left(n x^{3 n}\right)}}{x}\right) \left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) + \left(\frac{\operatorname{im}{\left(n x^{n}\right)}}{x} - \frac{3 \operatorname{im}{\left(n x^{3 n}\right)}}{x}\right) \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right)\right) \operatorname{sign}{\left(x^{3 n} - x^{n} \right)}}{x^{3 n} - x^{n}}
Simplificar
Segunda derivada [src] 
    /                                                                                                                                   /                           2                              2                                                                                                                                                                \                                                                                                                                                                    \
    |                                                                                                                                   |/      /   3*n\     /   n\\    /      /   3*n\     /   n\\    /    / 3*n\     / n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\   /    / 3*n\     / n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\|     / n /      2*n\\     /        2*n\ //    / 3*n\     / n\\ /      /   3*n\     /   n\\   /    / 3*n\     / n\\ /      /   3*n\     /   n\\\     / n /      2*n\\|
 -n |//    / 3*n\     / n\\ /      /   3*n\     /   n\\   /    / 3*n\     / n\\ /      /   3*n\     /   n\\\ d /    / n /      2*n\\\   \\- 3*im\n*x   / + im\n*x //  + \- 3*re\n*x   / + re\n*x //  - \- im\x   / + im\x //*\- im\n *x / - 3*im\n*x   / + 9*im\n *x   / + im\n*x // - \- re\x   / + re\x //*\- re\n *x / - 3*re\n*x   / + 9*re\n *x   / + re\n*x ///*sign\x *\-1 + x   //   n*\-1 + 3*x   /*\\- im\x   / + im\x //*\- 3*im\n*x   / + im\n*x // + \- re\x   / + re\x //*\- 3*re\n*x   / + re\n*x ///*sign\x *\-1 + x   //|
x  *|\\- im\x   / + im\x //*\- 3*im\n*x   / + im\n*x // + \- re\x   / + re\x //*\- 3*re\n*x   / + re\n*x ///*--\sign\x *\-1 + x   /// + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------|
    |                                                                                                        dx                                                                                                                                                 x                                                                                                                                                                                             /      2*n\                                                                |
    \                                                                                                                                                                                                                                                                                                                                                                                                                                                       x*\-1 + x   /                                                                /
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                                                                                                                                                                                                                                                                /      2*n\                                                                                                                                                                                                                                                               
                                                                                                                                                                                                                                                              x*\-1 + x   /
x−n(−n(3x2n−1)((re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n))+(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)))sign⁡(xn(x2n−1))x(x2n−1)+((re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n))+(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)))∂∂xsign⁡(xn(x2n−1))+(−(re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n)−re⁡(n2xn)+9re⁡(n2x3n))+(re⁡(nxn)−3re⁡(nx3n))2−(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)−im⁡(n2xn)+9im⁡(n2x3n))+(im⁡(nxn)−3im⁡(nx3n))2)sign⁡(xn(x2n−1))x)x(x2n−1)\frac{x^{- n} \left(- \frac{n \left(3 x^{2 n} - 1\right) \left(\left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)\right) \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x \left(x^{2 n} - 1\right)} + \left(\left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)\right) \frac{\partial}{\partial x} \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)} + \frac{\left(- \left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)} - \operatorname{re}{\left(n^{2} x^{n}\right)} + 9 \operatorname{re}{\left(n^{2} x^{3 n}\right)}\right) + \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right)^{2} - \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)} - \operatorname{im}{\left(n^{2} x^{n}\right)} + 9 \operatorname{im}{\left(n^{2} x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)^{2}\right) \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x}\right)}{x \left(x^{2 n} - 1\right)}
Simplificar
Tercera derivada [src] 
    /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   /                           2                              2                                                                                                                                                                \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     \
    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |/      /   3*n\     /   n\\    /      /   3*n\     /   n\\    /    / 3*n\     / n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\   /    / 3*n\     / n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\| d /    / n /      2*n\\\                                                                                                                                                                    /        2*n\ //    / 3*n\     / n\\ /      /   3*n\     /   n\\   /    / 3*n\     / n\\ /      /   3*n\     /   n\\\ d /    / n /      2*n\\\                     /                           2                              2                                                                                                                                                                \                                          2                                                                                                                             |
    |                                                                                                          2                         /  /    / 3*n\     / n\\ /       / 3  3*n\       /   3*n\       / 2  n\       /   n\        / 2  3*n\     / 3  n\\   /    / 3*n\     / n\\ /       / 3  3*n\       /   3*n\       / 2  n\       /   n\        / 2  3*n\     / 3  n\\     /      /   3*n\     /   n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\     /      /   3*n\     /   n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\\     / n /      2*n\\   2*\\- 3*im\n*x   / + im\n*x //  + \- 3*re\n*x   / + re\n*x //  - \- im\x   / + im\x //*\- im\n *x / - 3*im\n*x   / + 9*im\n *x   / + im\n*x // - \- re\x   / + re\x //*\- re\n *x / - 3*re\n*x   / + 9*re\n *x   / + re\n*x ///*--\sign\x *\-1 + x   ///     //    / 3*n\     / n\\ /      /   3*n\     /   n\\   /    / 3*n\     / n\\ /      /   3*n\     /   n\\\ /           2*n        2*n\     / n /      2*n\\   2*n*\-1 + 3*x   /*\\- im\x   / + im\x //*\- 3*im\n*x   / + im\n*x // + \- re\x   / + re\x //*\- 3*re\n*x   / + re\n*x ///*--\sign\x *\-1 + x   ///       /        2*n\ |/      /   3*n\     /   n\\    /      /   3*n\     /   n\\    /    / 3*n\     / n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\   /    / 3*n\     / n\\ /    / 2  n\       /   3*n\       / 2  3*n\     /   n\\|     / n /      2*n\\      2 /        2*n\  //    / 3*n\     / n\\ /      /   3*n\     /   n\\   /    / 3*n\     / n\\ /      /   3*n\     /   n\\\     / n /      2*n\\|
 -n |//    / 3*n\     / n\\ /      /   3*n\     /   n\\   /    / 3*n\     / n\\ /      /   3*n\     /   n\\\  d /    / n /      2*n\\\   \- \- im\x   / + im\x //*\- 27*im\n *x   / - 6*im\n*x   / - 3*im\n *x / + 2*im\n*x / + 27*im\n *x   / + im\n *x // - \- re\x   / + re\x //*\- 27*re\n *x   / - 6*re\n*x   / - 3*re\n *x / + 2*re\n*x / + 27*re\n *x   / + re\n *x // + 3*\- 3*im\n*x   / + im\n*x //*\- im\n *x / - 3*im\n*x   / + 9*im\n *x   / + im\n*x // + 3*\- 3*re\n*x   / + re\n*x //*\- re\n *x / - 3*re\n*x   / + 9*re\n *x   / + re\n*x ///*sign\x *\-1 + x   //                                                                                                                                                                                                                                   dx                         n*\\- im\x   / + im\x //*\- 3*im\n*x   / + im\n*x // + \- re\x   / + re\x //*\- 3*re\n*x   / + re\n*x ///*\1 - n - 3*x    + 9*n*x   /*sign\x *\-1 + x   //                                                                                                                             dx                         2*n*\-1 + 3*x   /*\\- 3*im\n*x   / + im\n*x //  + \- 3*re\n*x   / + re\n*x //  - \- im\x   / + im\x //*\- im\n *x / - 3*im\n*x   / + 9*im\n *x   / + im\n*x // - \- re\x   / + re\x //*\- re\n *x / - 3*re\n*x   / + 9*re\n *x   / + re\n*x ///*sign\x *\-1 + x   //   2*n *\-1 + 3*x   / *\\- im\x   / + im\x //*\- 3*im\n*x   / + im\n*x // + \- re\x   / + re\x //*\- 3*re\n*x   / + re\n*x ///*sign\x *\-1 + x   //|
x  *|\\- im\x   / + im\x //*\- 3*im\n*x   / + im\n*x // + \- re\x   / + re\x //*\- 3*re\n*x   / + re\n*x ///*---\sign\x *\-1 + x   /// - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------|
    |                                                                                                          2                                                                                                                                                                                                                                              2                                                                                                                                                                                                                                                                                                                                                  x                                                                                                                                                                                                      2 /      2*n\                                                                                                                                             /      2*n\                                                                                                                                                                                                  2 /      2*n\                                                                                                                                                                                                            2                                                                 |
    |                                                                                                        dx                                                                                                                                                                                                                                              x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         x *\-1 + x   /                                                                                                                                           x*\-1 + x   /                                                                                                                                                                                                 x *\-1 + x   /                                                                                                                                                                                               2 /      2*n\                                                                  |
    \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     x *\-1 + x   /                                                                  /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         /      2*n\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       x*\-1 + x   /
x−n(2n2(3x2n−1)2((re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n))+(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)))sign⁡(xn(x2n−1))x2(x2n−1)2−2n(3x2n−1)((re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n))+(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)))∂∂xsign⁡(xn(x2n−1))x(x2n−1)−2n(3x2n−1)(−(re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n)−re⁡(n2xn)+9re⁡(n2x3n))+(re⁡(nxn)−3re⁡(nx3n))2−(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)−im⁡(n2xn)+9im⁡(n2x3n))+(im⁡(nxn)−3im⁡(nx3n))2)sign⁡(xn(x2n−1))x2(x2n−1)−n((re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n))+(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)))(9nx2n−n−3x2n+1)sign⁡(xn(x2n−1))x2(x2n−1)+((re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n))+(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)))∂2∂x2sign⁡(xn(x2n−1))+2(−(re⁡(xn)−re⁡(x3n))(re⁡(nxn)−3re⁡(nx3n)−re⁡(n2xn)+9re⁡(n2x3n))+(re⁡(nxn)−3re⁡(nx3n))2−(im⁡(xn)−im⁡(x3n))(im⁡(nxn)−3im⁡(nx3n)−im⁡(n2xn)+9im⁡(n2x3n))+(im⁡(nxn)−3im⁡(nx3n))2)∂∂xsign⁡(xn(x2n−1))x−(−(re⁡(xn)−re⁡(x3n))(2re⁡(nxn)−6re⁡(nx3n)−3re⁡(n2xn)+27re⁡(n2x3n)+re⁡(n3xn)−27re⁡(n3x3n))+3(re⁡(nxn)−3re⁡(nx3n))(re⁡(nxn)−3re⁡(nx3n)−re⁡(n2xn)+9re⁡(n2x3n))−(im⁡(xn)−im⁡(x3n))(2im⁡(nxn)−6im⁡(nx3n)−3im⁡(n2xn)+27im⁡(n2x3n)+im⁡(n3xn)−27im⁡(n3x3n))+3(im⁡(nxn)−3im⁡(nx3n))(im⁡(nxn)−3im⁡(nx3n)−im⁡(n2xn)+9im⁡(n2x3n)))sign⁡(xn(x2n−1))x2)x(x2n−1)\frac{x^{- n} \left(\frac{2 n^{2} \left(3 x^{2 n} - 1\right)^{2} \left(\left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)\right) \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x^{2} \left(x^{2 n} - 1\right)^{2}} - \frac{2 n \left(3 x^{2 n} - 1\right) \left(\left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)\right) \frac{\partial}{\partial x} \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x \left(x^{2 n} - 1\right)} - \frac{2 n \left(3 x^{2 n} - 1\right) \left(- \left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)} - \operatorname{re}{\left(n^{2} x^{n}\right)} + 9 \operatorname{re}{\left(n^{2} x^{3 n}\right)}\right) + \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right)^{2} - \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)} - \operatorname{im}{\left(n^{2} x^{n}\right)} + 9 \operatorname{im}{\left(n^{2} x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)^{2}\right) \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x^{2} \left(x^{2 n} - 1\right)} - \frac{n \left(\left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)\right) \left(9 n x^{2 n} - n - 3 x^{2 n} + 1\right) \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x^{2} \left(x^{2 n} - 1\right)} + \left(\left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)\right) \frac{\partial^{2}}{\partial x^{2}} \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)} + \frac{2 \left(- \left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)} - \operatorname{re}{\left(n^{2} x^{n}\right)} + 9 \operatorname{re}{\left(n^{2} x^{3 n}\right)}\right) + \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right)^{2} - \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)} - \operatorname{im}{\left(n^{2} x^{n}\right)} + 9 \operatorname{im}{\left(n^{2} x^{3 n}\right)}\right) + \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right)^{2}\right) \frac{\partial}{\partial x} \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x} - \frac{\left(- \left(\operatorname{re}{\left(x^{n}\right)} - \operatorname{re}{\left(x^{3 n}\right)}\right) \left(2 \operatorname{re}{\left(n x^{n}\right)} - 6 \operatorname{re}{\left(n x^{3 n}\right)} - 3 \operatorname{re}{\left(n^{2} x^{n}\right)} + 27 \operatorname{re}{\left(n^{2} x^{3 n}\right)} + \operatorname{re}{\left(n^{3} x^{n}\right)} - 27 \operatorname{re}{\left(n^{3} x^{3 n}\right)}\right) + 3 \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)}\right) \left(\operatorname{re}{\left(n x^{n}\right)} - 3 \operatorname{re}{\left(n x^{3 n}\right)} - \operatorname{re}{\left(n^{2} x^{n}\right)} + 9 \operatorname{re}{\left(n^{2} x^{3 n}\right)}\right) - \left(\operatorname{im}{\left(x^{n}\right)} - \operatorname{im}{\left(x^{3 n}\right)}\right) \left(2 \operatorname{im}{\left(n x^{n}\right)} - 6 \operatorname{im}{\left(n x^{3 n}\right)} - 3 \operatorname{im}{\left(n^{2} x^{n}\right)} + 27 \operatorname{im}{\left(n^{2} x^{3 n}\right)} + \operatorname{im}{\left(n^{3} x^{n}\right)} - 27 \operatorname{im}{\left(n^{3} x^{3 n}\right)}\right) + 3 \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)}\right) \left(\operatorname{im}{\left(n x^{n}\right)} - 3 \operatorname{im}{\left(n x^{3 n}\right)} - \operatorname{im}{\left(n^{2} x^{n}\right)} + 9 \operatorname{im}{\left(n^{2} x^{3 n}\right)}\right)\right) \operatorname{sign}{\left(x^{n} \left(x^{2 n} - 1\right) \right)}}{x^{2}}\right)}{x \left(x^{2 n} - 1\right)}
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