Implicit Differentiation

Differentiating equations where y is not isolated as a function of x

Golden Rule: Differentiate both sides with respect to x. Every term involving y picks up a dy/dx factor from the chain rule. Then isolate dy/dx algebraically.

• Step 1: Differentiate both sides with respect to x — For pure x terms: use standard differentiation rules. For pure y terms: differentiate normally but multiply the result by dy/dx (chain rule). For mixed xy terms: use the product rule.
• Step 2: Collect all dy/dx terms on one side — Move all terms containing dy/dx to one side of the equation and all other terms to the other side.
• Step 3: Factor out dy/dx and solve — Factor dy/dx from the left side, then divide both sides by the remaining factor. The result is dy/dx expressed in terms of x and y (this is normal — do not try to eliminate y).