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Derivative Rules
Constant, Sum, Difference & Constant Multiple

Master the foundational differentiation rules that let you take derivatives of any polynomial: the constant rule, the sum and difference rules, and the constant multiple rule.

AP Calculus BC

What Is a Derivative?

The derivative of a function \( f(x) \) measures the instantaneous rate of change of \( f \) with respect to \( x \). Formally:

Computing this limit every time is slow. The derivative rules we learn today give us shortcuts that make differentiation fast and mechanical.

Quick Recall: The Power Rule

Before we dive in, recall the rule you already know for monomials:

\( x^3 \)
\( \frac{d}{dx}[x^3] = 3x^2 \)
\( x^{-2} \)
\( \frac{d}{dx}[x^{-2}] = -2x^{-3} \)
\( x^{1/2} \)
\( \frac{d}{dx}[x^{1/2}] = \tfrac{1}{2}x^{-1/2} \)

Today's rules let us combine power-rule results to differentiate any polynomial or linear combination of functions.

Why Do These Rules Matter?

\( f(x) \) slope = \( f'(x) \) \( x \) \( y \)

Real-world functions are rarely single terms: they are sums, differences, and scalar multiples of simpler pieces. Today's rules let you differentiate each piece independently and combine the results: no limit computation required.

Rule 1: The Constant Rule

\( f(x) = c \) \( \text{slope} = 0 \) everywhere \( x \) \( y \)

Intuition: A constant function is a horizontal line: it has zero slope at every point, so its derivative is 0.

Rule 2: The Constant Multiple Rule

In words: you can "pull out" any constant multiplier before differentiating. The constant just scales the rate of change.

Example 1
\( \frac{d}{dx}[5x^3] = 5 \cdot 3x^2 = 15x^2 \)
Example 2
\( \frac{d}{dx}[-7x^4] = -7 \cdot 4x^3 = -28x^3 \)

Rule 3: The Sum Rule

In words: the derivative of a sum is the sum of the derivatives. Differentiate each term separately, then add the results.

Rule 4: The Difference Rule

In words: the derivative of a difference is the difference of the derivatives. This follows directly from the sum rule (since subtraction is addition of a negative).

All Four Rules at a Glance

Rule Statement Quick Example
Constant \( \frac{d}{dx}[c] = 0 \) \( \frac{d}{dx}[7] = 0 \)
Constant Multiple \( \frac{d}{dx}[cf] = c \cdot f' \) \( \frac{d}{dx}[3x^2] = 6x \)
Sum \( \frac{d}{dx}[f+g] = f'+g' \) \( \frac{d}{dx}[x^2+x] = 2x+1 \)
Difference \( \frac{d}{dx}[f-g] = f'-g' \) \( \frac{d}{dx}[x^3-x] = 3x^2-1 \)

Sum/Difference vs. Product/Quotient

A common source of confusion: these rules apply to addition and subtraction only. They do not extend to products or quotients.

✓ Correct
\( \frac{d}{dx}[f + g] = f' + g' \)

The derivative of a sum splits term by term.
✕ Incorrect
\( \frac{d}{dx}[f \cdot g] \neq f' \cdot g' \)

Products require the product rule (a separate rule).

How to Apply the Rules

1
Rewrite the function so every term is in the form \( c \cdot x^n \). Expand products, split fractions, convert radicals to fractional exponents.
2
Apply the sum/difference rule: differentiate each term independently.
3
Apply the constant multiple rule: pull out any constant coefficient before differentiating.
4
Use the power rule on each remaining \( x^n \) term (and the constant rule on any standalone constants).
5
Simplify and combine like terms.

Example 1: Basic Polynomial

1
Split by sum/difference: \( \frac{d}{dx}[3x^4] - \frac{d}{dx}[5x^2] + \frac{d}{dx}[8x] - \frac{d}{dx}[12] \)
2
Pull out constants: \( 3 \cdot \frac{d}{dx}[x^4] - 5 \cdot \frac{d}{dx}[x^2] + 8 \cdot \frac{d}{dx}[x] - \frac{d}{dx}[12] \)
3
Power rule + constant rule: \( 3(4x^3) - 5(2x) + 8(1) - 0 \)
4
Simplify: \( 12x^3 - 10x + 8 \)

Example 2: Rewrite First

1
Rewrite: \( \frac{4}{x^2} = 4x^{-2} \) and \( 3\sqrt{x} = 3x^{1/2} \).
2
Differentiate each term:
\( \frac{d}{dx}[4x^{-2}] = 4(-2)x^{-3} = -8x^{-3} \)
\( \frac{d}{dx}[3x^{1/2}] = 3 \cdot \tfrac{1}{2}x^{-1/2} = \tfrac{3}{2}x^{-1/2} \)
3
Combine: \( -8x^{-3} - \tfrac{3}{2}x^{-1/2} = -\frac{8}{x^3} - \frac{3}{2\sqrt{x}} \)

Example 3: Visualizing the Derivative

Let \( f(x) = x^3 - 3x \). Then \( f'(x) = 3x^2 - 3 \). Let's see both graphs:

\( x \) \( y \) 1 -1 2 -2 \( f(x) = x^3 - 3x \) \( f'(x) = 3x^2 - 3 \)

Example 4: Combining All Rules

1
Sum/Difference rule: differentiate each of the four terms separately.
2
Constant multiple + Power rule:
\( 6 \cdot 5x^4 = 30x^4 \)
\( -\tfrac{2}{3} \cdot 3x^2 = -2x^2 \)
\( 4 \cdot \tfrac{1}{2}x^{-1/2} = 2x^{-1/2} \)
\( \frac{d}{dx}[\pi] = 0 \) (constant rule: \( \pi \) is just a number!)
3
Result: \( 30x^4 - 2x^2 + \frac{2}{\sqrt{x}} \)

Common Mistakes to Avoid

✕ Differentiating Products Term-by-Term
Writing \( \frac{d}{dx}[x^2 \cdot \sin x] = 2x \cdot \cos x \). The sum/difference rule does not apply to products: use the product rule instead.
✕ Forgetting that \( \pi, e \) Are Constants
\( \pi \) and \( e \) are numbers, not variables. \( \frac{d}{dx}[\pi] = 0 \) and \( \frac{d}{dx}[e] = 0 \). Only \( e^x \) has a special derivative.
✕ Dropping the Constant Multiplier
Writing \( \frac{d}{dx}[5x^3] = 3x^2 \) instead of \( 15x^2 \). The constant stays: multiply it through after applying the power rule.

Question 1 of 5

What is \( \frac{d}{dx}[7x^3 - 4x + 9] \)?

Question 2 of 5

Find \( \frac{d}{dx}\!\left[\frac{6}{x^3}\right] \).

Question 3 of 5

What is \( \frac{d}{dx}[5\sqrt{x} + 2x^2 - \pi] \)?

Question 4 of 5

If \( g(x) = -3x^4 + 2x^3 - x + 10 \), find \( g'(1) \).

Question 5 of 5

Which statement is true about the derivative rules we learned?