find dS/dA for S=2A(4-A)/[(1+A)^2]?

Answer 1

S = (8A - 2A^2) / (1 + A)^2

S = F(A) / G(A)

F(A) = 8A - 2A^2
F ' (A) = 8 - 4A

G(A) = (1 + A)^2
G ' (A) = 2(1 + A)(1) = 2(1 + A)

dS/dA = (F'G - G'F) / G^2
= [(8-4A)(1+A)^2 - (2)(1+A)(8A-2A^2)] / ((1 + A)^2)^2
= (4(2 - A)(1 + 2A + A^2) - (2 + 2A)(8A-2A^2)] / (1 + A)^4
= [4(2+4A+2A^2-A-2A^2-A^3) - (16A-4A^2+16A^2-4A^3] / (1 + A)^4
= [4(2+3A-A^3) - (16A+12A^2-4A^3)]/(1 + A)^4
= [8 + 12A - 4A^3 - 16A - 12A^2 + 4A^3] / (1 + A)^4
= (8 - 4A - 12A^2) / (1 + A)^4
= 4(2 - A - 3A^2) / (1 + A)^4
= 4(2 - 3A)(1 + A) / (1 + A)^4
= 4(2 - 3A) / (1 + A)^3

Answer 2

S = 2A(4-A)/(1+A)^2

dS/dA = 2(4-A)/(1+A)^2 - 2A/(1+A)^2 - 4A(4-A)/(1+A)^3

= 1/(1+A)^3*[ 2(4-A)(1+A) - 2A(1+A) -4A(4-A)]

= 1/(1+A)^3*[ 8-12A]