Question Detail

To find dy/dx, differentiate the given implicit equation x³+x+y³+3y=6 with respect to x:d/dx(x³) + d/dx(x) + d/dx(y³) + d/dx(3y) = d/dx(6)3x² + 1 + 3y²(dy/dx) + 3(dy/dx) = 0Now, group the terms containing dy/dx:(3y² + 3) dy/dx = -3x² - 1Solve for dy/dx:dy/dx = (-3x² - 1) / (3y² + 3)Finally, substitute the given point (x,y) = (1,1) into the expression for dy/dx:dy/dx = (-3(1)² - 1) / (3(1)² + 3)dy/dx = (-3 - 1) / (3 + 3)dy/dx = -4 / 6dy/dx = -2/3.This matches option B.