Differentiation from First Principles
Visualizing the slope of a curve using the limit concept
Explore how differentiation works by observing how the secant line approaches the tangent line as two points get closer together. Adjust the point distance to see the concept of limits in action!
Current Calculation:
First principles formula:
f'(x) = limh→0 [f(x+h) - f(x)] / h
Current slope: (f(1.5) - f(1)) / 0.5 = 2.5
The Concept of Differentiation from First Principles
Key Concepts:
Differentiation from first principles is the fundamental concept behind calculus:
- The derivative represents the instantaneous rate of change of a function
- It's found by calculating the slope of the secant line between two points
- As the distance h between points approaches zero, the secant becomes the tangent
- The process of taking the limit as h→0 gives us the exact slope at a point
Visual Interpretation:
The blue dashed line shows the tangent (exact derivative) while the green line shows the secant (approximation). As you move the slider to decrease h, watch how the green secant line approaches the blue tangent line.
