Overview
This lecture covers how to find derivatives of polynomial functions, introduces rules for differentiation (like the power and linearity rules), and discusses conditions for differentiability, especially at points of discontinuity, cusps, or corners.
Definition and Notation of Derivatives
- The derivative f'(x) is defined as the limit: limₕ→0 [f(x+h) - f(x)]/h.
- Geometrically, the derivative represents the slope of the tangent line to the curve at a point.
- Differentiation is the process of finding the derivative; a function is differentiable at x if this limit exists.
- Common notations: f'(x), y', dy/dx, d/dx[f(x)].
Basic Differentiation Rules
- The derivative of a constant is zero.
- The differential operator (d/dx) is linear: d/dx[cf(x)] = c·f'(x) and d/dx[f(x) ± g(x)] = f'(x) ± g'(x).
The Power Rule
- For f(x) = xⁿ (n positive integer), f'(x) = n·xⁿ⁻¹.
- The power rule also applies to negative and fractional exponents (e.g., 1/x and √x).
Differentiating Polynomials
- To differentiate a polynomial, apply the power rule term-by-term and sum the results.
- Example: d/dx[6x⁵ - 3x⁴ + 9x - 2] = 30x⁴ - 12x³ + 9.
Differentiability and Continuity
- A function must be continuous at x to be differentiable there.
- Not every continuous function is differentiable; points of non-differentiability can occur at corners, cusps, or vertical tangents.
Examples of Non-Differentiable Points
- Absolute value function (|x|): not differentiable at x=0 due to a corner.
- f(x) = x^(2/5): not differentiable at x=0 due to a cusp.
- f(x) = x^(1/3): not differentiable at x=0 due to a vertical tangent.
- f(x) = 1/x²: not differentiable at x=0 due to discontinuity.
Piecewise Functions
- For piecewise functions, check continuity and differentiability at the "join" point using limits and definitions.
- It's possible to construct piecewise functions that are both continuous and differentiable at the join.
Key Terms & Definitions
- Derivative (f'(x)) — Instantaneous rate of change of a function at point x.
- Differentiable — A function is differentiable at x if its derivative exists there.
- Power Rule — The derivative of xⁿ is n·xⁿ⁻¹.
- Linear Operator — An operator satisfying linearity: sum and scalar multiple rules.
- Continuous — A function without breaks or jumps at a given point.
Action Items / Next Steps
- Memorize the formal definition of the derivative.
- Practice applying the power, sum, and constant rules to differentiate polynomials.
- Review examples of points where functions are not differentiable.
- Prepare for test questions asking for the definition and computation of derivatives using the limit definition.