Implicit Derivative Calculator
Calculate derivatives of implicitly defined functions using chain rule and algebraic manipulation. Our calculator handles circles, ellipses, hyperbolas, and complex implicit equations with detailed step-by-step solutions and chain rule applications for comprehensive mathematical understanding.
Last updated: December 15, 2024
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Use ^ for exponents, = for equations. Examples: x^2 + y^2 = r^2, xy = c
Enter a point on the curve to evaluate the derivative numerically
Implicit Derivative Analysis
Original Equation:
Implicit Derivative (dy/dx):
Numerical Value at (x=3, y=4):
Slope of the tangent line at this point
Chain Rule Applications:
- d/dx(x²) = 2x
- d/dx(y²) = 2y(dy/dx)
- d/dx(25) = 0
Algebraic Manipulations:
- 2x + 2y(dy/dx) = 0
- 2y(dy/dx) = -2x
- dy/dx = -2x/(2y) = -x/y
Complete Solution Steps:
- Step 1: Start with the implicit equation: x^2 + y^2 = 25
- Step 2: Differentiate both sides with respect to x
- Step 3: Apply chain rule for terms involving y
- Step 4: Collect all terms with dy/dx
- Step 5: Solve for dy/dx
- Step 6: Evaluate at point (3, 4)
- Step 7: dy/dx = -0.75
Quick Example Result
For equation x² + y² = 25 at point (3,4):
dy/dx = -0.75
How This Calculator Works
Our implicit derivative calculator uses advanced differentiation techniques to find dy/dx for implicitly defined functions. The process involves applying the chain rule systematically, collecting terms with dy/dx, and solving algebraically to isolate the derivative. This approach works for complex equations where explicit solving for y would be difficult or impossible.
Implicit Differentiation Process
Step 1 - Differentiate Both Sides:
d/dx[F(x,y)] = d/dx[constant]Step 2 - Apply Chain Rule:
d/dx[y^n] = n·y^(n-1)·(dy/dx)Step 3 - Solve for dy/dx:
Collect terms and isolate dy/dxShows how the chain rule is applied to each term involving y
Mathematical Foundation
Implicit differentiation relies on the chain rule and the understanding that y is a function of x, even when not explicitly stated. When we differentiate y with respect to x, we get dy/dx. For more complex terms like y², we apply the chain rule: d/dx(y²) = 2y·(dy/dx). This systematic approach allows us to find slopes of curves defined by equations that cannot be easily solved for y.
- Chain rule application: Essential for differentiating composite functions involving y
- Algebraic manipulation: Collecting and isolating dy/dx terms requires careful algebra
- Function composition: Understanding that y = y(x) even in implicit form
- Geometric interpretation: dy/dx represents the slope of the tangent line to the curve
Sources & References
- Stewart Calculus - Implicit Differentiation and Related RatesComprehensive coverage of implicit differentiation techniques and applications
- MIT OpenCourseWare - Single Variable Calculus: Chain Rule and Implicit DifferentiationDetailed examples and problem-solving strategies
- Khan Academy - Implicit Differentiation Tutorial SeriesStep-by-step videos on chain rule applications in implicit functions
Need help with other differentiation techniques? Check out our derivative calculator and partial derivative calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Equation:
- Equation: x² + y² = 25
- Point: (3, 4)
- Find: dy/dx
Solution Steps:
- Differentiate: d/dx(x² + y²) = d/dx(25)
- Apply chain rule: 2x + 2y(dy/dx) = 0
- Solve for dy/dx: 2y(dy/dx) = -2x
- Simplify: dy/dx = -x/y
- At (3,4): dy/dx = -3/4 = -0.75
Result: dy/dx = -x/y, and at point (3,4): dy/dx = -0.75
This represents the slope of the tangent line to the circle at point (3,4).
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