Answer:
dy/dx = (2xy^7 - 4y) / (y - 6x^2y^6)
Explanation:
To find dy/dx by implicit differentiation, we differentiate both sides of the equation with respect to x.
Differentiating 4x + In y=x²y^6 with respect to x, we get:
4 + (1/y) dy/dx = 2xy^6 + 6x^2y^5 dy/dx
Now, we can isolate dy/dx by moving the terms that contain it to one side of the equation, and moving the rest to the other side:
(1/y - 6x^2y^5) dy/dx = 2xy^6 - 4
Dividing both sides by (1/y - 6x^2y^5), we get:
dy/dx = (2xy^6 - 4) / (1/y - 6x^2y^5)
Simplifying this expression, we can write:
dy/dx = (2xy^7 - 4y) / (y - 6x^2y^6)