Topic 2.6 Notes – Derivative Rules: Constant, Sum, Difference, and Constant Multiple

The Four Linearity Rules

All four rules come from one big idea: derivatives respect addition, subtraction, and constant multiples. You don’t have to re-derive anything from limits each time. You use structure.

1. Constant Rule

If

$$f(x) = c$$

where $c$ is a number, then

$$f'(x) = 0.$$

A constant has no change. Its graph is a horizontal line, and horizontal lines have slope 0 everywhere.

Examples:

• $\frac{d}{dx}(8) = 0$
• $\frac{d}{dx}(-3) = 0$

2. Constant Multiple Rule

If

$$f(x) = c \cdot g(x),$$

then

$$f'(x) = c \cdot g'(x).$$

The constant just “comes along for the ride.”

Example:

• $\frac{d}{dx}(7x^3) = 7 \cdot \frac{d}{dx}(x^3) = 7(3x^2) = 21x^2$

You’ll use this constantly with coefficients in polynomials.

3. Sum Rule

If

$$f(x) = g(x) + h(x),$$

then

$$f'(x) = g'(x) + h'(x).$$

You differentiate each term separately and keep the plus sign.

4. Difference Rule

If

$$f(x) = g(x) - h(x),$$

then

$$f'(x) = g'(x) - h'(x).$$

Same idea, but keep the minus sign.

The Big Picture: Linearity

All four rules combine into one clean statement:

$$\frac{d}{dx}[a g(x) + b h(x)] = a g'(x) + b h'(x).$$

This is why differentiation of polynomials feels mechanical. You just apply the rule term by term.

Combining These with the Power Rule

You already know the power rule:

$$\frac{d}{dx}(x^n) = n x^{n-1}.$$

Polynomials are just sums and differences of constant multiples of powers of $x$. So every polynomial derivative is just:

• Constant multiple rule
• Power rule
• Sum/difference rule

all stacked together.

Differentiating a Polynomial

Take:

$$f(x) = 4x^5 - 3x^3 + 6x - 11.$$

Break it into terms and differentiate each one.

$$f'(x) = 20x^4 - 9x^2 + 6 - 0$$

Final answer:

$$f'(x) = 20x^4 - 9x^2 + 6.$$

Notice:

• The constant $-11$ vanished.
• The linear term $6x$ became 6.
• Signs stayed exactly as they were.

That “keep the sign, multiply by the exponent, drop the exponent by 1” rhythm should feel automatic.

Every term shifts down one degree. The constant term disappears.

When These Rules Apply

You’re safe using only these rules when:

• The function is a polynomial.
• Terms are added or subtracted.
• Coefficients are constants.
• There are no products like $(x+2)(x-5)$ that need product rule.
• There are no compositions like $(3x+1)^4$ that need chain rule.

On many no-calculator multiple-choice questions, the derivative is just a polynomial. If you’re slow here, you burn time you can’t afford.

Common Mistakes

• Forgetting constants go to 0.
• Dropping a negative sign when differentiating.
• Writing $\frac{d}{dx}(x^4) = 4x^4$ instead of $4x^3$.
• Turning $5x$ into 0 instead of 5.
• Using product rule when the function is already expanded.

Most errors on this topic are algebra errors, not calculus errors.

Why This Matters Later

These rules are the foundation for:

• Finding critical points
• Writing tangent lines
• Analyzing increasing/decreasing behavior
• Solving optimization problems

If polynomial derivatives aren’t automatic, everything else in Unit 3 becomes harder.