Given an equation of two variables x and y that might be difficult (or impossible?) to solve for y. y as an implicit function of x; "embedded" with x.
Can differentiate without solving for y. Differentiate both sides of the equation with respect to x; each y becoming dy/dx, or y', then solve for y'.
Now have a "formula" for the derivative.

Example differentiating:
(y³)' = 3y²y'
(x+3y)' = 1+3y'
(xy²)' = 1y² + x2yy' = y² + 2xyy'

Example equations:
x³+xy = 7-y³
(x³+xy)' = (7-y³)'
3x²+ y+xy' = 3y²y'
3x²+ y = 3y²y'-xy'
y'= (3x²+y) / (3y²-x)

sin y = x
(sin y)' = (x)'
cos y y' = 1
y' = 1 / cos y = ... = 1 / √(1-x²)