Calculus Mastery

The Human Knowledge Project

Chapter 6 — Basic Differentiation Rules

Power Rule, Constant Rule, Sum Rule, and Constant Multiple Rule

6.1 Learning Objectives

By the end of this chapter, students should be able to:

Differentiate basic polynomial functions.
Apply the Power Rule correctly.
Differentiate constants.
Differentiate sums and differences of functions.

Apply the Constant Multiple Rule.

Interpret derivatives geometrically and physically.

Understand why differentiation rules simplify calculus enormously.

Visualize derivatives as slope functions.

Explain patterns connecting powers and derivatives.

Build fluency with basic derivative computation.

6.2 Big Picture — Why Differentiation Rules Matter

In Chapter 5, students computed derivatives from the definition:

′

(x)=

h→0

lim

f(x+h)−f(x)

This method is foundational.

But it is also:

slow

algebraically heavy

impractical for large problems

Imagine computing derivatives of:

or:

−4x

+12x

−3

using only the limit definition every time.

The algebra would become enormous.

Fortunately, mathematicians discovered patterns.

These patterns became:

differentiation rules

The rules allow derivatives to be computed rapidly while preserving the deeper meaning:

slopes

rates of change

local behavior

6.3 The Power Rule — One of the Most Important Rules in Calculus

Suppose:

f(x)= x

where:

n is a positive integer

Then:

)= nx

n−1

This is called:

the Power Rule

6.4 Understanding the Pattern

Students should not merely memorize this rule.

They should SEE the pattern.

Examples:

(x)=1

)=2x

)=3x

)=4x

Notice the pattern:

exponent moves downward

exponent decreases by one

This pattern becomes one of the central computational tools in calculus.

6.5 Why the Power Rule Makes Sense Graphically

Consider:

f(x)= x

Earlier we discovered:

′

(x)=2x

Interpretation:

slope changes depending on x

At:

x = 1 → slope = 2

x = 5 → slope = 10

The graph becomes:

steeper

as x increases.

Now consider:

f(x)= x

Its derivative:

shows slope increases even faster.

Students should visualize:

higher powers grow increasingly steep

derivatives measure how rapidly steepness changes

6.6 Worked Example — Applying the Power Rule

Find derivative of:

f(x)= x

Apply rule:

bring exponent downward

subtract 1 from exponent

Result:

′

(x)=5x

6.7 Another Example

Differentiate:

f(x)= x

Derivative:

′

(x)=7x

6.8 The Constant Rule

Suppose:

f(x)=5

This function never changes.

Its graph is horizontal.

Horizontal lines have slope:

Therefore:

(c)=0

for any constant:

6.9 Why Constants Have Zero Derivative

Students should think physically.

If:

temperature never changes

position never changes

height never changes

then:

rate of change equals zero

No change means:

zero derivative

6.10 Worked Example — Constant Rule

Differentiate:

f(x)=12

Derivative:

′

(x)=0

6.11 The Constant Multiple Rule

Suppose:

f(x)=7x

The constant:

simply scales the function.

The derivative becomes:

′

(x)=7(3x

)

=21x

6.12 Why This Rule Makes Sense

Multiplying a function by:

makes:

all slopes 7 times steeper

The derivative scales naturally.

Students should visualize:

graph stretched vertically

tangent slopes stretched similarly

6.13 Worked Example — Constant Multiple Rule

Differentiate:

f(x)=4x

Derivative:

′

(x)=24x

6.14 The Sum Rule

Suppose:

f(x)= x

The derivative becomes:

′

(x)=3x

+2x

The derivative distributes across addition.

6.15 Why the Sum Rule Makes Sense

Suppose:

one function contributes one type of change

another contributes another type

Total change equals:

sum of their individual changes

This is physically intuitive.

6.16 Worked Example — Sum Rule

Differentiate:

f(x)= x

+3x

Differentiate term-by-term:

)=4x

(3x

)=6x

(7)=0

Combine:

′

(x)=4x

+6x

6.17 The Difference Rule

Subtraction behaves similarly.

Example:

f(x)= x

−x

Derivative:

′

(x)=5x

−2x

6.18 Polynomial Differentiation

Students now possess enough tools to differentiate many polynomials.

Example:

f(x)=3x

−2x

+7x

−9

Differentiate term-by-term:

′

(x)=15x

−6x

+14x

Constant disappears because:

(−9)=0

6.19 Visualization — Derivative as a New Function

This is a major conceptual leap.

The derivative itself is another function.

Original function:

f(x)

Derivative:

′

(x)

At every x-value:

derivative gives tangent slope

Students should visualize:

one graph describing height

another graph describing steepness

6.20 Positive and Negative Derivatives

Suppose:

′

(x)>0

Then:

graph rising

Suppose:

′

(x)<0

Then:

graph falling

Suppose:

′

(x)=0

Then:

graph flat horizontally

This becomes extremely important later in:

optimization

curve sketching

maxima/minima

6.21 Physical Interpretation

Suppose:

s(t)= t

represents position.

Then:

′

(t)=3t

represents velocity.

At:

larger t-values

velocity increases rapidly

The object accelerates.

Derivatives allow us to describe motion precisely.

6.22 Why Differentiation Rules Matter Historically

Before calculus:

complicated changing systems were difficult to analyze

Differentiation rules transformed mathematics.

Scientists could suddenly analyze:

motion

astronomy

engineering

fluid flow

optimization

with unprecedented power.

These rules became one of the great revolutions in science.

6.23 Common Student Mistakes

Mistake 1 — Forgetting to Reduce Exponent

Incorrect:

)=4x

Correct:

Mistake 2 — Differentiating Constants Incorrectly

Constants differentiate to:

Mistake 3 — Algebra Errors

Students often simplify incorrectly after differentiating.

Careful algebra still matters.

Mistake 4 — Memorizing Without Understanding

Students should remember:

derivatives describe rates

derivatives describe slopes

differentiation measures local change

The rules are shortcuts built from the limit definition.

6.24 Problem-Solving Strategy

When differentiating:

Step 1

Identify function type.

Step 2

Differentiate term-by-term.

Step 3

Apply:

Power Rule

Constant Rule

Sum/Difference Rules

Constant Multiple Rule

Step 4

Simplify carefully.

Step 5

Interpret result:

graphically

physically

6.25 Practice Problems

A. Basic Power Rule

)

Explain pattern in the Power Rule.

B. Constant Rule

(7)

(−3)

(π)

Explain why constants have derivative zero.

Describe graph of constant function.

C. Constant Multiple Rule

(5x

)

(9x

)

(−2x

)

Explain how vertical stretching affects slopes.

Explain physical meaning of scaling derivatives.

D. Sum and Difference Rules

+3x

)

−x

)

(3x

+2x

−7)

(7x

−4x

+2x−9)

Explain why derivatives distribute across addition.

E. Graphical Interpretation

Explain what positive derivative means graphically.

Explain what negative derivative means graphically.

Explain what zero derivative means graphically.

Describe how steepness changes for:

F. Physical Interpretation

Suppose:

s(t)= t

Find velocity function.

Suppose:

s(t)= t

Find velocity function.

Explain why velocity changes faster for higher powers.

Explain why derivatives are useful in motion.

Explain why derivatives measure responsiveness.

6.26 Selected Solutions

Problem 2

)

Apply Power Rule:

=5x

Problem 6

(7)

Constants never change.

Derivative:

Problem 16

+3x

)

Differentiate term-by-term:

+6x

Problem 19

(7x

−4x

+2x−9)

Differentiate each term:

42x

−12x

Problem 26

Suppose:

s(t)= t

Velocity:

′

(t)=2t

6.27 Chapter Summary

In this chapter we introduced:

Power Rule

Constant Rule

Constant Multiple Rule

Sum Rule

Difference Rule

Most importantly:

students learned that differentiation rules provide efficient ways to compute:

slopes

rates of change

local behavior

without repeatedly using the full limit definition.

These rules become the foundation for nearly all future differentiation techniques.