Implicit Differentiation Calculator
Differentiate implicit equations and find dy/dx for curves defined by relations like x² + y² = r²
Examples: x^2 + y^2 (circle), xy (hyperbola), x^3 + y^3 - 3xy
結果
値を入力して計算をクリックして結果を表示してください。
Theory & Formula
What is Implicit Differentiation?
Implicit differentiation is a technique used to find derivatives of equations where y is not explicitly solved for. Instead of y = f(x), we have F(x,y) = 0.
Method
1. Differentiate both sides with respect to x
2. Treat y as a function of x and apply chain rule: d/dx[y^n] = n·y^(n-1)·dy/dx
3. Collect all dy/dx terms on one side
4. Solve for dy/dx
calculators.calculus.implicitDifferentiation.theory.example
Find dy/dx for the circle x² + y² = 25:
\(x^2 + y^2 = 25\)\(\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)\)\(2x + 2y\frac{dy}{dx} = 0\)\(\frac{dy}{dx} = -\frac{x}{y}\)
[calculus.implicitDifferentiation.theory.chainRule]
When differentiating terms with y, remember that y is a function of x, so you must apply the chain rule:
\(\frac{d}{dx}(y^n) = ny^{n-1}\frac{dy}{dx}\)