Section 3.8: Implicit Differentiation

Explicit Form vs. Implicit Form

• Explicit Form: Functions expressed as y = f(x).
  • Example: y = x^2 + 2x + 1.
  • Derivative found using the power rule: dy/dx = 2x + 2.
• Implicit Form: Functions not explicitly solved for y.
  • Example: x^2 + y^2 = xy.
  • Difficult or impossible to solve for y explicitly, requiring implicit differentiation.

Implicit Differentiation

• Purpose: Find dy/dx when y is not isolated in terms of x.
• Key Steps: Differentiation Process
  • Differentiate both sides of the equation with respect to x.
  • Apply the chain rule to y terms (since y is a function of x).
  • Solve for dy/dx.

Example 1: Circle Equation

• Given: x^2 + y^2 = 1.
• This represents a circle of radius 1 centered at the origin.
• Differentiate:
  • Left side: 2x + 2y(dy/dx) = 0.
  • Solve for dy/dx: dy/dx = -x/y.

Example 2: Another Equation

• Given: y - x^2 + 2x = 1.
• Implicit differentiation results in the same derivative as explicit differentiation.

Example 3: Complex Equation

• Given: y^2 + 3x = 2.
• Find derivative at the point (-1, √5).
• Solve: dy/dx = -3/(2y).
• At the point (-1, √5), the slope becomes -3/(2√5).

Complex Example with Product Rule

• Given: yx^2 + y^3 = cos(y).
• Use implicit and product rules:
  • Differentiate and solve: dy/dx = -2xy / (x^2 + 3y^2 + sin(y)).

Example with Exponential Function

• Given: x^2y^2 + e^y = tan(y).
• Implicit differentiation: dy/dx = -2xy^2 / (x^2y + e^y - sec^2(y)).

Application: Normal Line

• Given: x^3 + x^2y + 4y^2 = 6.
• Find normal line at point (1, 1).
• Slope of tangent at (1, 1) is -5/9; normal line slope is 9/5.

Additional Example

• Given: 1 + x = sin(xy^2).
• Using implicit differentiation, solve for dy/dx.

Intersection of Curves

• Hyperbola: x^2 - y^2 = 5.
• Ellipse: 4x^2 + 9y^2 = 72.
• Intersection point (3, 2) shows lines are perpendicular (slopes are negative reciprocals).

Caution with Implicit Equations

• Not every implicit equation represents a meaningful function.
• Example: e^(x^2 + y^2) = 0 has no solutions, hence its derivative is meaningless.
• Always consider if a function satisfying the equation exists.

Summary

• Implicit differentiation is crucial for equations not explicitly solvable for y.
• Be aware of assumptions concerning the existence of functions for implicit equations.