Overview
This lecture introduces the formal definition of the derivative, explains what it measures, and demonstrates how to compute derivatives using the power rule for different types of functions.
Understanding the Derivative
- The derivative at a point gives the slope of the tangent line at that point.
- The derivative represents the rate of change of a function at a specific input.
- A positive derivative indicates the function is increasing; a negative derivative means it's decreasing.
- The value of a function and its rate of change at a point are independent concepts.
Formal Definition
- The slope of a secant line between points (a, f(a)) and (x, f(x)) is [f(x) - f(a)] / (x - a).
- Letting x approach a, the secant slope approaches the tangent slope, defining the derivative.
- Alternatively, write the second point as (a + h, f(a + h)), and take the limit as h approaches zero.
- The derivative f'(a) is lim(h→0) [f(a + h) - f(a)] / h.
Derivatives of Basic Functions
- For a constant function f(x) = k, the derivative is zero everywhere: f'(x) = 0.
- For f(x) = x, the derivative is one: f'(x) = 1.
- For f(x) = x², the derivative is two times x: f'(x) = 2x.
- For f(x) = x³, the derivative is three times x squared: f'(x) = 3x².
The Power Rule
- The power rule states: the derivative of xⁿ is n·xⁿ⁻¹ for any positive integer n.
- The power rule enables quick computation of derivatives without using the limit definition each time.
Key Terms & Definitions
- Derivative — The slope of the tangent line to a function at a point; also the function’s rate of change at that point.
- Secant line — A line passing through two points on a curve.
- Tangent line — A line that touches a curve at one point with the same slope as the curve at that point.
- Power rule — A rule for differentiation: d/dx(xⁿ) = n·xⁿ⁻¹.
Action Items / Next Steps
- Practice using the power rule to find derivatives of various polynomial functions.