Introduction
Derivative: measures instantaneous rate of change of function output relative to input. Core concept in calculus. Enables analysis of dynamic systems, motion, growth, and optimization. Foundation for differential equations and mathematical modeling.
"The derivative measures how a function changes as its input changes." -- James Stewart
Conceptual Overview
Rate of Change
Derivative quantifies how rapidly output varies with respect to input. Unlike average rate, derivative captures instantaneous behavior.
Slope of Tangent Line
Derivative at a point equals slope of tangent line to graph at that point. Tangent approximates function locally.
Local Linearity
Functions differentiable at a point behave locally like linear functions. Derivative represents best linear approximation.
Formal Definition
Limit Definition
Derivative of function f at point x defined as limit:
f'(x) = limh→0 [f(x + h) - f(x)] / hExistence of Limit
Derivative exists if above limit exists and finite. Non-existence implies non-differentiability.
Domain Considerations
Derivative defined only at interior points of domain where limit can be evaluated.
Geometric Interpretation
Tangent Line Slope
Derivative equals slope of line tangent to curve y = f(x) at given x.
Secant to Tangent Transition
Secant line slope approaches tangent slope as interval shrinks.
Graphical Visualization
Derivative indicates steepness and direction of curve at specific points.
| Concept | Description |
|---|---|
| Secant Line | Line through two points on curve; average rate of change. |
| Tangent Line | Line touching curve at one point; instantaneous rate. |
Notation
Leibniz Notation
Expressed as dy/dx, indicating derivative of y with respect to x.
Lagrange Notation
Denoted f'(x), emphasizing function and point of differentiation.
Newton Notation
Used primarily in physics: ẏ represents time derivative of y.
Differentiability
Definition
Function differentiable at x if derivative exists there.
Continuity vs Differentiability
Differentiability implies continuity, but converse false.
Points of Non-Differentiability
Occurs at cusps, corners, vertical tangents, or discontinuities.
Properties of Derivatives
Linearity
D(af + bg) = aD(f) + bD(g), for constants a,b.
Product Rule
D(fg) = f'Dg + g'DfQuotient Rule
D(f/g) = (gDf - fDg) / g²Chain Rule
D(f(g(x))) = f'(g(x)) * g'(x)Derivatives of Common Functions
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ (n ≠ 0) | n xⁿ⁻¹ |
| sin x | cos x |
| cos x | -sin x |
| eˣ | eˣ |
| ln x (x > 0) | 1/x |
Power Rule
Most fundamental differentiation rule for polynomial functions.
Exponential and Logarithmic Rules
Derivatives maintain functional forms with specific multiplicative factors.
Applications
Physics
Velocity = derivative of position; acceleration = derivative of velocity.
Economics
Marginal cost, marginal revenue computed via derivatives.
Optimization
Critical points found via zero derivative to locate maxima/minima.
Engineering
Analyzing rates of change in systems, control theory.
Computation Techniques
Direct Application of Limit Definition
Used for foundational understanding, rarely practical for complex functions.
Rules of Differentiation
Product, quotient, chain rules simplify calculation.
Implicit Differentiation
Derivatives of functions defined implicitly rather than explicitly.
Higher-Order Derivatives
Repeated differentiation yields second, third, nth derivatives.
Limitations and Extensions
Non-Differentiable Functions
Functions with discontinuities, sharp points lack derivatives.
Generalizations
Directional derivatives, partial derivatives extend concept to multivariate functions.
Distributional Derivatives
Generalized derivatives in weak sense for non-classical functions.
Examples
Example 1: Polynomial
f(x) = 3x² + 5x - 4f'(x) = 6x + 5Example 2: Trigonometric
f(x) = sin xf'(x) = cos xExample 3: Exponential
f(x) = eˣf'(x) = eˣExample 4: Using Chain Rule
f(x) = (2x + 3)⁵f'(x) = 5(2x + 3)⁴ * 2 = 10(2x + 3)⁴References
- Stewart, J. Calculus: Early Transcendentals, Brooks/Cole, 8th ed., 2015, pp. 100-150.
- Apostol, T. M. Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, Wiley, 2nd ed., 1967, pp. 120-170.
- Spivak, M. Calculus, Publish or Perish, 4th ed., 2008, pp. 85-130.
- Thomas, G. B., Finney, R. L. Calculus and Analytic Geometry, Addison-Wesley, 9th ed., 1996, pp. 98-140.
- Rudin, W. Principles of Mathematical Analysis, McGraw-Hill, 3rd ed., 1976, pp. 130-160.