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Implicit differentiation finds dy/dx when y is not explicitly solved for x — when x and y appear together in an equation like x² + y² = 25 (a circle). Differentiate both sides with respect to x, treating y as a function of x and applying the chain rule whenever y appears.
When differentiating y with respect to x, each y term picks up a factor of dy/dx by the chain rule. After differentiating, collect all dy/dx terms on one side and solve algebraically.
Worked example: find dy/dx for x² + y² = 25. Differentiate: 2x + 2y(dy/dx) = 0. Solve: dy/dx = −x/y. This gives the slope of the tangent at any point (x,y) on the circle. At (3,4): dy/dx = −3/4.
Implicit differentiation handles curves that cannot be expressed as y = f(x) or that are easier to leave in implicit form, such as ellipses, folium of Descartes, and many level curves.
Differentiate both sides w.r.t. x; d/dx[y] = dy/dx (chain rule)
Choose a curve
Equation
x² + y² = 25
Evaluate slope at a specific point (x₀, y₀)
Key rule
d/dx[y] = dy/dx (y is a function of x — chain rule)
d/dx[y²] = 2y · dy/dx
d/dx[xy] = y + x · dy/dx (product + chain)
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.