To find y' (or dy/dx) through Implicit Differentiation, we need to use the Chain Rule when we see y. While the derivative of x with respect to x is 1, the derivative of y will be y' (or dy/dx) by the Chain Rule. This is where the "respect to" really needs to be observed.
x⁶ + y⁴ = -7
6x⁵ + y⁴ = -7 by taking the derivative of x⁶, using the Power Rule
6x⁵ + 4y³y' = -7 by taking the derivative IMPLICITLY of y⁴ and using the chain rule
6x⁵ + 4y³y' = 0 By taking the derivative of 0
Now we use algebra to get y' alone on one side, and not y' on the other.
6x⁵ + 4y³y' = 0
4y³y' = - 6x⁵
y' = -6x⁵ / 4y³
y' = -3x⁵ / 2y³.
