Calculus•Beginner

Understanding Derivatives

Learn how derivatives measure instantaneous rate of change and visualize tangent lines to understand the geometric interpretation of differentiation.

Problem Statement

Given a function f(x) = x², find the derivative at a point x = a and visualize the tangent line at that point.

Example:

Input: f(x) = x², a = 2

Output: f'(2) = 4

Tangent line: y = 4x - 4

Intuition

The derivative represents the instantaneous rate of change of a function. Geometrically, it's the slope of the tangent line to the curve at a specific point.

Key Concept: While the average rate of change uses two points, the derivative captures the rate of change at a single instant by taking the limit as the interval approaches zero.

For f(x) = x², the derivative isf'(x) = 2x. At x = 2, the slope is 4, meaning the function is increasing at a rate of 4 units vertically for every 1 unit horizontally.

Interactive Visualization

Drag the point along the curve to see how the derivative (tangent line slope) changes.

f(2.00) = 4.00f'(2.00) = 4.00
xy-2-11202468f(x) = x²slope = 4.00
Function
f(x) = x²
Point Value
f(2.00) = 4.00
Derivative (Slope)
f'(2.00) = 4.00

Try it: Drag the red point along the curve to see how the tangent line slope changes. Notice that the derivative f'(x) = 2x gives the exact slope at each point.

Implementation

derivative.py
import numpy as np
import matplotlib.pyplot as plt

def derivative(f, x, h=1e-5):
    """
    Calculate derivative using numerical differentiation
    f'(x) ≈ (f(x + h) - f(x - h)) / (2h)
    """
    return (f(x + h) - f(x - h)) / (2 * h)

def visualize_derivative(f, x_point):
    # Create x values
    x = np.linspace(x_point - 2, x_point + 2, 100)
    y = f(x)
    
    # Calculate derivative at point
    slope = derivative(f, x_point)
    
    # Tangent line: y = slope * (x - x_point) + f(x_point)
    tangent_y = slope * (x - x_point) + f(x_point)
    
    # Plot
    plt.plot(x, y, label='f(x)')
    plt.plot(x, tangent_y, label=f'Tangent (slope={slope:.2f})')
    plt.scatter([x_point], [f(x_point)], color='red', zorder=5)
    plt.legend()
    plt.grid(True)
    plt.show()

# Example: f(x) = x²
f = lambda x: x**2
visualize_derivative(f, x_point=2)  # Derivative at x=2 is 4

Computational Complexity

Time Complexity

O(1) - Analytical derivative

Using calculus rules, derivative is computed instantly

Space Complexity

O(1) - Constant space

Only stores the result value

Note: Numerical differentiation (as shown in code) requires O(1) function evaluations but may have precision errors. Symbolic differentiation using calculus rules is exact.

Related Topics

Chain RuleComing Soon

Derivatives of composite functions

IntegrationComing Soon

The inverse operation of differentiation

OptimizationComing Soon

Finding maxima and minima using derivatives

Taylor SeriesComing Soon

Approximating functions using derivatives

Discussion