1. Why Implicit Differentiation Is Needed
Implicit differentiation is used when a function is not written in the form y = f(x). Many relationships between x and y appear as equations where separating y is difficult or impossible. Instead of solving for y first, implicit differentiation allows differentiation directly from the given equation.
2. How Implicit Differentiation Works
The idea is simple: differentiate both sides of the equation with respect to x, treating y as a function of x. Whenever y is differentiated, multiply by dy/dx because y changes with x.
2.1. Key Rule
Whenever you differentiate y with respect to x, use:
\frac{d}{dx}(y) = \frac{dy}{dx}
2.2. Example of the Rule
\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}
3. Step-by-Step Procedure
- Differentiate both sides with respect to x.
- Apply the chain rule whenever y appears.
- Collect all terms with dy/dx on one side.
- Solve for dy/dx.
4. Examples
These examples show where implicit differentiation is most helpful.
4.1. Example 1 — Circle Equation
Differentiate:
x^2 + y^2 = 25
Differentiating both sides:
2x + 2y \frac{dy}{dx} = 0
Solving:
\frac{dy}{dx} = -\frac{x}{y}
4.2. Example 2 — Mixed Powers
Differentiate:
x^3 + x y + y^2 = 7
Differentiating term by term:
3x^2 + (x \frac{dy}{dx} + y) + 2y \frac{dy}{dx} = 0
Group dy/dx terms:
(x + 2y)\frac{dy}{dx} = -3x^2 - y
Final result:
\frac{dy}{dx} = \frac{-3x^2 - y}{x + 2y}
4.3. Example 3 — Trigonometric
Differentiate:
\sin(x + y) = x^2
Differentiate both sides:
\cos(x + y)(1 + \frac{dy}{dx}) = 2x
Solving for dy/dx:
\frac{dy}{dx} = \frac{2x}{\cos(x + y)} - 1
5. When Implicit Differentiation Is Useful
Implicit differentiation is essential for curves like circles, ellipses, and other equations where isolating y would be complicated. It also appears naturally in related rates and in functions defined by constraints.