Implicit Differentiation
Implicit differentiation is a technique for finding the derivative dy/dx when a relationship between x and y is defined implicitly by an equation, rather than expressed as y = f(x) explicitly. It involves differentiating both sides of the equation with respect to x and applying the chain rule whenever y is differentiated, since y is a function of x. This method is essential for finding slopes of curves defined by circles, ellipses, and other relations that cannot be easily solved for y.
Key Formula
dy/dx = −(∂F/∂x) / (∂F/∂y)
LaTeX: \frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}
| Symbol | Meaning | Unit |
|---|---|---|
| F(x, y) | implicit equation written as F(x,y) = 0 | dimensionless |
| ∂F/∂x | partial derivative of F with respect to x | dimensionless |
| ∂F/∂y | partial derivative of F with respect to y | dimensionless |
| dy/dx | derivative of y with respect to x | dimensionless |
Worked Example
Problem
Find dy/dx for the circle x² + y² = 25.
Solution
Step 1: Differentiate both sides with respect to x. Step 2: d/dx[x²] + d/dx[y²] = d/dx[25]. Step 3: Apply chain rule to y²: 2x + 2y·(dy/dx) = 0. Step 4: Solve for dy/dx: 2y·(dy/dx) = −2x. Step 5: dy/dx = −2x / (2y) = −x/y.
Answer
dy/dx = −x/y (valid wherever y ≠ 0 on the circle x² + y² = 25)
Implicit Differentiation for Common Curves
| Curve | Implicit Equation | dy/dx (implicitly) | Geometric Meaning |
|---|---|---|---|
| Circle | x² + y² = r² | −x/y | Slope of tangent to circle |
| Ellipse | x²/a² + y²/b² = 1 | −b²x / (a²y) | Tangent slope on ellipse |
| Folium of Descartes | x³ + y³ = 3axy | (ay − x²) / (y² − ax) | Tangent at any point |
| Implicit cubic | y³ + xy = 4 | −y / (3y² + x) | Tangent slope at any point |
Interactive Tools
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Related Terms
Chain Rule
The chain rule is a differentiation rule used to compute the derivative of a composite function, stating that the derivative of f(g(x)) equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. It is one of the most widely applied rules in calculus, essential whenever a function is "nested" inside another. The chain rule is critical in physics for related rates problems, in machine learning for backpropagation, and in multivariable calculus for total derivatives.
Related Rates
Related rates problems use implicit differentiation with respect to time to find how the rate of change of one quantity relates to the rate of change of another quantity, given that both are functions of time. A geometric or physical relationship between the quantities is differentiated with respect to time, and known rates are substituted to find the unknown rate. These problems are extensively used in physics and engineering, for instance to relate the rate at which a ladder slides down a wall to the rate at which the base moves away from the wall.
Differentiation
Differentiation is the process of computing the derivative of a function, yielding a new function that expresses the rate of change of the original at every point in its domain. It involves applying systematic rules — such as the power rule, product rule, chain rule, and quotient rule — to transform a given function into its derivative. Differentiation is used extensively in physics for velocity and acceleration, in economics for marginal analysis, and in engineering for optimisation and control systems.
The word "implicit" comes from the Latin "implicitus" meaning entwined or folded in. Implicit differentiation as a systematic technique was developed during the 17th century by Leibniz and Newton as part of their work on curves defined by polynomial equations.