Project 7: The Derivative Explorer (Calculus Visualization)

Build a tool that approximates derivatives and visualizes slope.

Project Overview

Attribute Value
Difficulty Level 2: Intermediate
Time Estimate Weekend
Main Language Python
Alternative Languages JavaScript, C++
Knowledge Area Calculus
Tools Plotting tool
Main Book “Calculus” by James Stewart

What you’ll build: A tool that plots a function alongside its numerical derivative and highlights slope at a point.

Why it teaches math: Derivatives are the language of change. Seeing slope evolve makes the concept real.

Core challenges you’ll face:

  • Approximating derivatives numerically
  • Balancing step size and accuracy
  • Visualizing tangent lines

Real World Outcome

You will input a function and see its derivative graph and tangent lines at chosen points.

Example Output:

$ python derivative.py "x**3 - 2*x" --point 1.5
f(1.5) = 0.375
f'(1.5) = 4.75
Saved plot to derivative.png

Verification steps:

  • Compare with analytical derivatives
  • Test multiple functions

The Core Question You’re Answering

“How can I measure how fast something changes at a single point?”

This project makes the idea of slope precise.

Concepts You Must Understand First

Stop and research these before coding:

  1. Limit definition of derivative
    • Why does the slope come from a limit of secant lines?
    • Book Reference: “Calculus” by James Stewart, Ch. 2
  2. Numerical differentiation
    • What is the difference between forward and central difference?
    • Book Reference: “Numerical Methods for Engineers” by Chapra & Canale, Ch. 4
  3. Tangent lines
    • How do you write the equation of a tangent line?
    • Book Reference: “Calculus” by James Stewart, Ch. 2

Questions to Guide Your Design

  1. Step size selection
    • How do you choose delta-x to balance error?
    • How will you avoid numerical noise?
  2. Visualization
    • Will you plot derivative on same axes or separate?
    • How will you show tangent lines clearly?

Thinking Exercise

Slope Approximation

Approximate the derivative of f(x) = x^2 at x = 3 using delta-x = 0.1 and delta-x = 0.01.

Questions while working:

  • Which approximation is closer?
  • Why can extremely small delta-x be unstable?

The Interview Questions They’ll Ask

Prepare to answer these:

  1. “What is the derivative in geometric terms?”
  2. “Why does the limit definition matter?”
  3. “What is the difference between forward and central difference?”
  4. “How do you draw a tangent line?”
  5. “Why does numerical differentiation amplify noise?”

Hints in Layers

Hint 1: Starting Point Start with a simple polynomial function.

Hint 2: Next Level Implement central difference for better accuracy.

Hint 3: Technical Details Expose delta-x as a parameter so you can test stability.

Hint 4: Tools/Debugging Compare results to symbolic derivatives for validation.

Books That Will Help

Topic Book Chapter
Derivative basics “Calculus” by James Stewart Ch. 2
Numerical differentiation “Numerical Methods for Engineers” by Chapra & Canale Ch. 4
Tangent lines “Calculus” by James Stewart Ch. 2

Implementation Hints

  • Keep evaluation safe by restricting function inputs.
  • Use central difference for better accuracy.
  • Label tangent lines clearly on the plot.

Learning Milestones

  1. First milestone: You can approximate derivatives numerically.
  2. Second milestone: You can visualize tangent lines correctly.
  3. Final milestone: You can explain numerical errors and stability.