For an equation f(x, y) = c, differentiating both sides with respect to x gives: df/dx + df/dy * (dy/dx) = 0. Solving for dy/dx yields the formula below. This works for any differentiable implicit equation.
The formula dy/dx = -(df/dx) / (df/dy) directly uses partial derivatives of f with respect to each variable. This is the same operation computed by the Partial Derivative Calculator, which finds df/dx and df/dy separately for any polynomial in x and y.
To find the second derivative d²y/dx² of an implicitly defined function, differentiate dy/dx with respect to x. Since dy/dx = -(df/dx) / (df/dy), apply the quotient rule to this expression, then substitute your earlier expression for dy/dx.
The key step is substituting the expression for dy/dx back into the result after applying the quotient rule. This eliminates dy/dx from the second derivative expression and leaves d²y/dx² in terms of x and y only. For the reverse operation of differentiation, see the Antiderivative Calculator.
Implicit differentiation and partial derivatives are deeply connected. The formula dy/dx = -(df/dx) / (df/dy) can be read as: the ratio of the partial derivative of f with respect to x, divided by the partial derivative of f with respect to y, negated.
| Operation | What it computes | Notation | Key rule applied |
|---|---|---|---|
| Partial df/dx | Rate of change w.r.t. x, y held constant | df/dx or f_x | Power rule, y treated as constant |
| Partial df/dy | Rate of change w.r.t. y, x held constant | df/dy or f_y | Power rule, x treated as constant |
| Implicit dy/dx | How y changes as x changes on the curve | dy/dx | Chain rule, dy/dx factor on y terms |
| Relationship | dy/dx = -(df/dx) / (df/dy) | Implicit Function Theorem | Both operations used together |
The Implicit Function Theorem guarantees that as long as df/dy is not zero at a point, the implicit equation F(x, y) = c locally defines y as a differentiable function of x near that point. For polynomial estimation between known data points (a related numerical technique), see the Interpolation Calculator.
Equation: x² + y² = 25. Find dy/dx at (3, 4).
The slope of the tangent to the circle x² + y² = 25 at point (3, 4) is -0.75. This is consistent with the geometric fact that the tangent is perpendicular to the radius from the origin to (3, 4), which has slope 4/3.