Implicit Differentiation

Ever wonder how to find derivatives when you can't solve for y? Implicit differentiation is your solution! This technique works by differentiating both sides of an equation with respect to x while treating y as a function of x.

The process is straightforward: Start with an equation in the form F(x,y)=0, differentiate both sides with respect to x (remembering that y is a function of x), then solve for dy/dx. When differentiating terms with y, you'll need to use the chain rule by multiplying by dy/dx.

For example, with x²+y²=1, we differentiate to get 2x + 2y(dy/dx) = 0, then solve for dy/dx to find dy/dx = -x/y. For more complex equations like xy + y² = 4, we apply the product rule where needed, giving us dy/dx = -y/$x+2y$.

Pro Tip: When differentiating terms like y², remember to use the chain rule: d/dx(y²) = 2y(dy/dx). This is the most common mistake students make with implicit differentiation!