Stellar Implicit differentiation is used when a relation is given by an equation involving both x and y, rather than by solving directly for y as a function of x. Instead of rewriting the equation first, we differentiate both sides with respect to x and treat y as a function of x. This means that whenever we differentiate a y term, we must multiply by by the chain rule. Implicit differentiation is important because many curves, such as circles and ellipses, are not naturally expressed as y= one function of x, yet we still want to find slopes and rates of change on them.
The key idea is that y depends on x, even if that dependence is not written explicitly. So, for example, the derivative of with respect to x is not just 2y; it is . You must stay alert for this every time a y appears. After differentiating, the goal is usually to solve algebraically for dy/dx. This process lets us find tangent slopes and derivative formulas even when solving for y first would be difficult or messy.