Implicit Differentiation?

Question

Find the slope of the tangent line to the curve

√(12x+2y) + √(3xy)= 21.4

at the point (6,8)

Answer 1

(12x + 2y)^(1/2) + (3xy)^(1/2) = 21.4

[(1/2)(12x + 2y)^(-1/2)][12 + 2dy] + [(1/2)(3xy)^(-1/2)][3x dy + 3y] = 0

[(1/2)(12(6) + 2(8))^(-1/2)][12 + 2dy] + [(1/2)(3(6)(8))^(-1/2)][3(6) dy + 3(8)] = 0

[(1/2)(72 + 16)^(-1/2)][12 + 2dy] + [(1/2)(144)^(-1/2)][18 dy + 24] = 0

[(1/2)(88)^(-1/2)][12 + 2dy] + [(1/2)(1/12)][18 dy + 24] = 0

[(1/2)(1/(4sqrt(11)))][12 + 2dy] + [1/24][18 dy + 24] = 0

[(1/8sqrt(11))][12 + 2dy] + [3dy/4 + 1] = 0

[(3/2sqrt(11)) + (dy/4sqrt(11))] + [3dy/4 + 1] = 0

dy/4sqrt(11) + 3dy/4 = -1 - (3/2sqrt(11))

dy[1/4sqrt(11) + 3/4] = -1 - (3/2sqrt(11))

dy[(1+ 3sqrt(11))/4sqrt(11)] = -1 - (3/2sqrt(11))

dy = [-1 - (3/2sqrt(11))] [4sqrt(11)/(1+ 3sqrt(11))]

Basically this boils down to taking the derivative of everything with respect to x. You need some chain and product rules here, but beyond that, it's simplification - getting the dy term by itself.