Given p = 100 - 6x2
we know that R = px
R = $(100 - 6x^2)x = 100x - 6x^3$
Marginal Revenue (MR) = $"dR"/"dx" = "d"/"dx" (100x - 6x^3)$
LHS = MR = $100 - 18x^2$ ....(1)
Differentiating p, with respect to, 'x' we get,
$"dp"/"dx" = - 12 x$
$"dx"/"dp" = (-1)/(12x)$
$therefore eta_"d" = - "p"/x * "dx"/"dp"$
$= (- (100 - 6x^2))/x * ((-1)/(12x))$
$= (100 - 6x^2)/(12x^2)$
$therefore "RHS" = "p"(1-1/(eta_"d"))$
$= (100 - 6x^2)(1 - (12x^2)/(100 - 6x^2))$
RHS = $(cancel(100 - 6x^2))((100 - 6x^2 - 12x^2)/cancel(100 - 6x^2))$
= 100 - 18x2 ...(2)
From (1) and (2), LHS = RHS
$therefore "MR" = "p"(1 - 1/eta_"d")$
