Final answer:
To find y/x by implicit differentiation, differentiate both sides of the equation with respect to x and isolate dy/dx. The resulting expression is dy/dx = 12x / (1 - 6y).
Step-by-step explanation:
To find y/x by implicit differentiation, we start by differentiating both sides of the equation with respect to x. Using the chain rule, we have:
d/dx (y) = d/dx (6x² + 3y²)
On the left side, we have dy/dx, and on the right side, we differentiate each term separately:
dy/dx = 12x + 6y(dy/dx)
Next, we isolate dy/dx by moving the terms involving dy/dx to one side:
dy/dx - 6y(dy/dx) = 12x
Factoring out dy/dx gives:
(1 - 6y)dy/dx = 12x
Finally, we solve for dy/dx by dividing both sides by 1 - 6y:
dy/dx = 12x / (1 - 6y)