Study Notes on Implicit Differentiation, Derivatives, and Limits

Key Concepts of Implicit Differentiation and Derivatives

Defining Functions

• Explicit vs. Implicit Definitions

  • Example: y = x² + 3x - 4 defines y explicitly.

  • Rearranging to x² + 3x - y = 4 defines y implicitly.

Derivatives of Functions

• For y = x² + 3x - 4:

  • Derivative: y' = 2x + 3 (explicit differentiation).

• Applying Implicit Differentiation:

  • For x² + 3x - y = 4:

  • Derivative:

    • d/dx of x² is 2x.

    • d/dx of 3x is 3.

    • d/dx of y is y'.

    • d/dx of 4 is 0.

  • Collecting terms:

  • y' = 2x + 3.

Working with Circular Functions

• For x² + y² = 4:

  • Solve for y: y = ±√(4 - x²).

  • Positive root covers the upper half-circle; negative root covers the lower half.

  • Derivative (implicit differentiation):

  • 2x + 2yy' = 0.

  • Solve for y': y' = -x/y.

• Example Values:

  • If x = 1, then y = ±√3 → corresponding points (1, √3) and (1, -√3).

Second Derivatives

• To find y'':

  • Derivative of y' = -x/y.

  • Use quotient rule:

  • Apply to original function or explicit form, results in the same derivative.

  • Simplified second derivative: y'' = -4/(y²).

General Principles

• Remember to apply the chain rule for every instance of y:

  • If y = f(x), then dy/dx = f'(x)y'.

• Formulas & Connections:

  • y' obtained through implicit differentiation should correlate with explicit derivative results, even if not visually identical.

Practical Applications

• Example Problem: x + x²y³ = 4y + 1

  • Find the tangent line at the point (1, 2): Calculate slope y' using implicit differentiation.

  • y' = -17/8: Write the tangent line equation.

Absolute Value Inequalities

• Handle using interval notation:

  • Solve |4x - 3| ≤ 11.

  • Result is -2 ≤ x ≤ 7/2.

Quadratic Inequalities

• Solve by transforming to zero:

  • Setup x² + 6x - 7 < 0 to factor or apply quadratic formula, determining ranges.

Continuity of Functions

• Definition involves three conditions:

  1. Function defined at c.

  2. The limit exists at c.

  3. Limit equals function value at c.

• Example: Identify removable discontinuities (fixable by redefining functions).

Limits and Derivatives

• Limit Definitions:

  1. For polynomial functions, limits often straightforward.

  2. Use L'Hôpital's rule judiciously in indeterminate forms (0/0).

  3. Apply transformation to recognize behaviors as x approaches certain points.

Conclusion

• Review implicit differentiation and limit processes to be prepared for exams.