Calculus: Implicit Differentiation and Related Rates

Introduction

• Discusses the peculiarities encountered while learning calculus, particularly with circles and tangents.
• Example: Circle with radius 5 centered at the origin (Equation: x² + y² = 5²).
• Objective: Find the slope of the tangent line to the circle at point (3, 4).

Key Concepts

Tangent Line to Circle

• The tangent line is perpendicular to the radius at the point of tangency.
• The curve is not the graph of a function; we can't use simple derivatives.
• Introduces implicit curves: sets of all points (x, y) satisfying a specific equation.
• Implicit differentiation steps:
  • Derivative of x²: 2x · dx
  • Derivative of y²: 2y · dy
  • Rearrange to solve for dy/dx: dy/dx = -x/y
  • Example at (3, 4): Slope = -3/4

Implicit Differentiation

• Why implicit differentiation feels strange:
  • Derivatives include dy and dx parts.
  • Conceptualize tiny steps (dx, dy) along the curve.
  • Ensure the curve remains unchanged by these tiny steps.

Related Rates Problem

• Example: Ladder problem
  • Ladder length = 5m
  • Distance from top of ladder to ground initially = 4m
  • Distance from bottom to wall = 3m
  • Top of the ladder drops at 1 m/s.

Steps for Solution

1. Label distances as functions of time: y(t) and x(t)
2. Use Pythagorean theorem: x(t)² + y(t)² = 5²
3. Calculate derivatives:
   • dy/dt = -1 m/s
   • Apply chain rule:
     • Derivative of x² = 2x · dx/dt
     • Derivative of y² = 2y · dy/dt
4. Solve for dx/dt: Result = 4/3 m/s

Comparison to Implicit Differentiation

• Example: Circle tangent line vs. ladder problem
• Both involve the same equation x² + y² = 5²
• Derivatives interpreted in terms of time for the ladder problem but differently for the circle.

Exploring Derivative Interpretation

• Use the function s(x, y) = x² + y²
• Derivative ds represents change in s due to tiny steps (dx, dy)
• Important to keep ds = 0 for steps along the curve

New Example: Function sin(x · y²) = x

• Take derivative using product rule and chain rule:
  • Left side: sin(x) · 2y · dy + y² · cos(x) · dx
  • Right side: dx
• Set expressions equal, solve for dy/dx

Example: Natural Log Function

• y = ln(x)
• Differentiate to find slope:
  • Rearrange to e^y = x
  • Derivative: e^y · dy = dx
  • Solve for dy/dx: dy/dx = 1/x

Multivariable Calculus Connection

• Functions with multiple inputs and their changes.
• Clear understanding of tiny nudges and their interdependencies.

Conclusion

• Next topic: Limits and formalizing the concept of a derivative.