If y = ( ax+b cx+d ) , then 2 dy dx d^3 y dx^3 is equal to

Step-by-step solution

Given y = ax+b cx+d Differentiating with respect to x , we get dy dx = a(cx+d) - c(ax+b) (cx+d)^2 = ad-bc (cx+d)^2 Let ad-bc = k . Then dy dx = k(cx+d)⁻² Differentiating again with respect to x , we get d^2 y dx^2 = -2kc(cx+d)⁻³ Differentiating once more with respect to x , we get d^3 y dx^3 = 6kc^2(cx+d)⁻⁴ Evaluating 2 dy dx d^3 y dx^3 : 2 dy dx d^3 y dx^3 = 2 ( k(cx+d)⁻² ) ( 6kc^2(cx+d)⁻⁴ ) = 12k^2 c^2 (cx+d)⁻⁶ Evaluating 3 ( d^2 y dx^2 )^2 : 3 ( d^2 y dx^2 )^2 = 3 ( -2kc(cx+d)⁻³ )^2 = 12k^2 c^2 (cx+d)⁻⁶ Therefor