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

Derivative Rules for Linear Combinations of Functions

A linear combination just means functions connected by addition, subtraction, or constant multiples. Derivatives behave nicely with these operations.

Constant Rule

If $f(x) = c$, where $c$ is a number,

$$f'(x) = 0$$

A constant has no change. Graphically, it’s a horizontal line with slope 0, so its derivative is 0 everywhere.

Constant function and its zero derivative

On quizzes, students often accidentally carry the constant down. Don’t. It disappears.

Sum Rule

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

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

The derivative distributes across addition. You differentiate each piece separately and keep the plus sign.

Difference Rule

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

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

Same idea as the sum rule. The minus sign stays attached to the term. Losing that negative is one of the most common mistakes on tests.

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: If $f(x) = 7x^3$, then $f'(x) = 7 \cdot 3x^2 = 21x^2$

You multiply the coefficient by the exponent. Don’t just lower the exponent.

Combining These with the Power Rule

Most of the time, you’ll use these rules together with the power rule:

If $f(x) = x^n$, then $$f'(x) = n x^{n-1}$$

Polynomials are just sums of constant multiples of powers of $x$. That’s why this topic is foundational.

A general polynomial:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

Its derivative:

$$f'(x) = n a_n x^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1$$

The constant term $a_0$ vanishes.

Differentiating Polynomial Functions

When you see only powers of $x$ connected by + or −, you’re in this topic. No product rule. No chain rule.

Example:

$$f(x) = 4x^5 - 3x^3 + 8x - 6$$

Differentiate term by term:

• $4x^5 \rightarrow 20x^4$
• $-3x^3 \rightarrow -9x^2$
• $8x \rightarrow 8$
• $-6 \rightarrow 0$

So,

$$f'(x) = 20x^4 - 9x^2 + 8$$

That’s the entire process.

Structural Pattern to Recognize

Any polynomial fits this general structure:

General polynomial structure and term-by-term differentiation

Your brain should automatically think: “Differentiate each term separately and keep the signs.”

This is especially helpful on AP multiple choice where speed matters. Polynomial derivatives should take seconds.

Common Errors That Cost Easy Points

• Forgetting to multiply coefficients
  $5x^4 \rightarrow 20x^3$, not $5x^3$
• Dropping negatives
  $-2x^6 \rightarrow -12x^5$
• Messing up exponents
  Exponent decreases by 1.
  If exponent is 1, you get a constant.
  If exponent is 0, derivative becomes 0.
• Overcomplicating
  If terms are added, do not use product rule. Rewrite first if needed.

Why This Matters for BC

These rules are the backbone for:

• Finding critical points
• Analyzing graphs
• Setting up differential equations
• Taylor and Maclaurin polynomials later in BC

If you can differentiate polynomials instantly and accurately, harder problems become manageable because this step won’t slow you down.