Introduction to Differential Calculus
Introduction to the fundamental concepts of differential calculus
Interactive Differential Calculus Visualization
Welcome to the Differential Calculus Explorer
This interactive visualization helps you understand the fundamental concepts of differential calculus: derivatives, tangent lines, and secant lines. Explore how secant lines approach tangent lines as the distance between points approaches zero, which is the core idea behind derivatives.
What you can explore:
- Polynomial functions - Learn with f(x) = x² and see how derivatives work
- Trigonometric functions - Explore f(x) = sin(x) and its derivative
- Exponential functions - Discover f(x) = e^x and its unique property
- Rational functions - Understand f(x) = 1/x and its derivative
How to Use This Visualization
Interactive Features:
- • Select Function Type - Choose from 4 different functions to explore
- • Adjust Point x - Use the slider or click on the graph to set the point
- • Adjust Distance h - Control the distance between two points on the curve
- • Toggle Visualizations - Show/hide secant line, tangent line, and derivative value
- • Animate h → 0 - Watch the secant line approach the tangent line
What You'll See:
- • Function Curve (solid, colored) - The original function f(x)
- • Secant Line (purple, dashed) - Line connecting two points on the curve
- • Tangent Line (red) - Line that touches the curve at exactly one point
- • Main Point (colored circle) - The point where the tangent is drawn
- • Second Point (purple circle) - The second point forming the secant line
- • Derivative Value - Real-time display of f'(x) at the selected point
- • Calculation Steps - Step-by-step breakdown of the derivative calculation
Click on the graph to move the point
As h → 0, secant line approaches tangent line
Current Values
Point: (1.000, 1.000)
Second Point: (2.000, 4.000)
Secant Slope: 3.0000
Tangent Slope (f'): 2.0000
Key Concepts
Derivative: The slope of the tangent line at a point represents the instantaneous rate of change.
Secant Line: A line connecting two points on the curve. As the distance h approaches 0, the secant line approaches the tangent line.
Tangent Line: A line that touches the curve at exactly one point, representing the derivative.
Derivative Definition
f'(x) = limh→0 [f(x+h) - f(x)] / h
The derivative is the limit of the difference quotient as h approaches zero. This is what we visualize when the secant line approaches the tangent line.
Lessons
Individual learning units
Lessons coming soon