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age: derivative(abs(xn-a*N/(1-exp(-i*2*pi*k/N)))^2,a)
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8*(xn + 8*a/(e^(-1/4*I*pi*k) - 1))/(e^(1/4*I*pi*k) - 1) + 8*(8*a/(e^(1/4*I*pi*k) - 1) + conjugate(xn))/(e^(-1/4*I*pi*k) - 1)

age: derivative(abs(xn-a*N/(1-exp(-i*2*pi*k/N)))^2,a) ....: ....: ....: 8*(xn
sage: derivative(abs(xn - aN/(1 - exp(-i2pik/N)))^2, a) 8(xn + 8*a/(e^(-1/4*I*pi*k) 8a/(e^(-1/4Ipik) - 1))/(e^(1/4*I*pi*k) 1))/(e^(1/4Ipik) - 1) + 8*(8*a/(e^(1/4*I*pi*k) 8(8a/(e^(1/4Ipik) - 1) + conjugate(xn))/(e^(-1/4*I*pi*k) conjugate(xn))/(e^(-1/4Ipik) - 1)
1)

Try this:

sage: derivative(abs(xn - aN/(1 a*N/(1 - exp(-i2pik/N)))^2, exp(-i*2*pi*k/N)))^2, a)
    8(xn 8*(xn + 8a/(e^(-1/4Ipik) 8*a/(e^(-1/4*I*pi*k) - 1))/(e^(1/4Ipik) 1))/(e^(1/4*I*pi*k) - 1) + 8(8a/(e^(1/4Ipik) 8*(8*a/(e^(1/4*I*pi*k) - 1) + conjugate(xn))/(e^(-1/4Ipik) conjugate(xn))/(e^(-1/4*I*pi*k) - 1)1)

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