Question

If the derivative dx/dy (instead of dy/dx ) exists at a point and dx/dy =0 , then the tangent line at that point is vertical. Calculate dx/dy for the equation y⁴+8=3x²+3y². (Use symbolic notation and fractions where needed. )

dx/dy = ____

Answer 1

To find dx/dy for the given equation, differentiate both sides with respect to y, isolate dx/dy, and simplify the equation.

Step-by-step explanation:

To find dx/dy for the equation y⁴+8=3x²+3y², we need to differentiate both sides of the equation with respect to y. This will give us the derivative dy/dy on the left side, and the derivative dx/dy on the right side. Differentiating the equation, we get:

4y³ * (dy/dy) + 0 = 6x * (dx/dy) + 6y * (dy/dy)

Since dy/dy is simply 1, we can simplify the equation to:

4y³ = 6x * (dx/dy) + 6y

Now, we can isolate dx/dy by moving the other terms to the opposite side of the equation:

6x * (dx/dy) = 4y³ - 6y

Finally, we can solve for dx/dy by dividing both sides of the equation by 6x:

dx/dy = (4y³ - 6y) / (6x)