Term by Term Differentiation

Run the Term by Term Differentiation MicroSim Fullscreen

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Description

This MicroSim demonstrates the step-by-step process of differentiating polynomials by showing how each term is individually differentiated using the power rule, and then combined using the sum rule. The visualization makes the abstract process of differentiation concrete by displaying:

Original Terms: Each term of the polynomial displayed in its own row
Rules Applied: The specific differentiation rule used for each term (Power Rule, Constant Rule)
Results: The derivative of each individual term
Final Answer: The complete derivative assembled from all individual term results

How to Use

Select a Polynomial: Use the dropdown menu to choose from preset polynomials of varying complexity
Step Through: Click "Next Step" to see each term differentiated one at a time, with highlighting to show the current term being processed
Show All: Click "Show All" to instantly reveal all differentiation steps and the final answer
Reset: Click "Reset" to start over with the same polynomial
Toggle Graph View: Check "Show Graph" to see the original function f(x) in blue and its derivative f'(x) in green plotted together

Rules Demonstrated

Rule	Description	Example
Power Rule	d/dx(x^n) = nx^(n-1)	d/dx(x^4) = 4x^3
Constant Multiple Rule	d/dx(cf(x)) = c * f'(x)	d/dx(3x^4) = 3 * 4x^3 = 12x^3
Constant Rule	d/dx(c) = 0	d/dx(7) = 0
Sum/Difference Rule	d/dx(f + g) = f' + g'	Combines all term derivatives

Delta Moment

"See how each term gets its own makeover? The power rule is like a recipe: bring down the exponent, multiply, then subtract one from the power. Do that for each term, add them all up, and you have your derivative!"

Lesson Plan

Learning Objectives

By the end of this activity, students will be able to:

Apply the power rule to differentiate polynomial terms (Bloom Level 3)
Calculate derivatives of polynomials by applying the sum/difference rule (Bloom Level 3)
Execute the step-by-step procedure for polynomial differentiation (Bloom Level 3)

Target Audience

AP Calculus AB/BC students
High school students (grades 11-12)
College students in Calculus I

Prerequisites

Understanding of polynomial functions
Knowledge of exponent rules
Introduction to the concept of derivatives

Activity: Predict-Then-Reveal

Time: 15-20 minutes

Warm-up (3 min): Review the power rule formula: d/dx(x^n) = nx^(n-1)
Prediction Phase (5 min):
Display the polynomial 3x^4 - 2x^2 + 5x - 7
Have students write down their prediction for each term's derivative
Create a table in their notes matching the MicroSim layout
Verification Phase (7 min):
Use "Next Step" to reveal each derivative one at a time
Students check their predictions and correct any errors
Discuss any surprises (especially the constant term becoming 0)
Independent Practice (5 min):
Students select different presets and predict before revealing
Toggle "Show Graph" to visualize the relationship between f(x) and f'(x)

Assessment

Formative: Monitor student predictions during the activity
Summative: Provide 3-5 polynomials for students to differentiate without the MicroSim, then verify with the tool

Extension Activities

Have students create their own polynomial and manually calculate the derivative, then verify using the graph view
Discuss patterns: What happens to the degree of a polynomial when you differentiate?
Challenge: Predict where f'(x) = 0 based on the graph of f(x)

References

Power Rule - Khan Academy - Comprehensive review of the power rule with examples and practice problems
Derivative Rules - Paul's Online Math Notes - Complete reference for basic differentiation formulas including power rule and constant multiple rule
p5.js Reference - Documentation for the p5.js library used to create this MicroSim