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Derivative Rules
Constant, Sum, Difference & Constant Multiple

Master the foundational rules for computing derivatives: how to differentiate constants, sums, differences, and constant multiples of functions.

AP Calculus AB

What Is a Derivative?

A derivative measures how a function changes as its input changes. Formally, the derivative of \( f(x) \) is defined as:

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

Visualizing the Tangent Line

x y f(x) tangent (a, f(a)) slope = f'(a)

The derivative \( f'(a) \) gives the slope of the tangent line to \( f(x) \) at the point \( (a, f(a)) \). Instead of computing limits every time, we use derivative rules as shortcuts.

Why Do Rules Matter?

Computing derivatives from the limit definition every time is tedious and error-prone. Derivative rules let you differentiate quickly and reliably by recognizing patterns.

Efficiency
Skip the limit computation entirely for common function forms.
Building Blocks
These rules combine to differentiate any polynomial or linear combination.
Foundation
Product, quotient, and chain rules all build on these basics.

The Constant Rule

If \( f(x) = c \) where \( c \) is a constant, then:

$$ \frac{d}{dx}[c] = 0 $$

x y f(x) = c slope = 0 everywhere

The Constant Multiple Rule

If \( c \) is a constant and \( f(x) \) is differentiable, then:

$$ \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) $$

Example: \( \frac{d}{dx}[5x^3] = 5 \cdot \frac{d}{dx}[x^3] = 5 \cdot 3x^2 = 15x^2 \)

The Sum Rule

If \( f(x) \) and \( g(x) \) are both differentiable, then:

$$ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $$

Example: \( \frac{d}{dx}[x^2 + x^4] = 2x + 4x^3 \)

The Difference Rule

If \( f(x) \) and \( g(x) \) are both differentiable, then:

$$ \frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x) $$

Example: \( \frac{d}{dx}[x^5 - x^3] = 5x^4 - 3x^2 \)

Sum vs. Difference at a Glance

Rule Statement In Words
Sum \( (f+g)' = f' + g' \) Derivative of sum = sum of derivatives
Difference \( (f-g)' = f' - g' \) Derivative of difference = difference of derivatives
Constant Multiple \( (cf)' = c \cdot f' \) Constants factor out of derivatives
Constant \( (c)' = 0 \) Derivative of any constant is zero

Combining Rules Visually

x y 1 2
\( f(x) = x^2 \)
\( g(x) = 3x \)
\( h(x) = x^2 + 3x \)
 h'(x) = f'(x) + g'(x) h'(x) = 2x + 3 Sum rule in action

When \( h(x) = f(x) + g(x) \), the slope of \( h \) at every point equals the sum of the slopes of \( f \) and \( g \). The sum rule captures this graphical fact algebraically.

How to Apply the Rules

1
Identify each term. Break the expression into individual terms separated by \( + \) or \( - \).
2
Factor out constants. For each term like \( c \cdot f(x) \), pull \( c \) outside the derivative.
3
Differentiate each term. Apply the power rule or recognize constants: \( \frac{d}{dx}[x^n] = nx^{n-1} \).
4
Reassemble. Add or subtract the differentiated terms following the original signs.
5
Simplify. Combine like terms and clean up coefficients.

Example 1: Polynomial

Find \( \frac{d}{dx}[3x^4 + 7x^2 - 5] \).

1
Split by sum/difference rule: \( \frac{d}{dx}[3x^4] + \frac{d}{dx}[7x^2] - \frac{d}{dx}[5] \)
2
Pull out constants: \( 3 \cdot \frac{d}{dx}[x^4] + 7 \cdot \frac{d}{dx}[x^2] - 0 \)
3
Power rule: \( 3 \cdot 4x^3 + 7 \cdot 2x - 0 \)
4
Simplify: \( 12x^3 + 14x \)

Example 2: Negative & Fractional Powers

Find \( \frac{d}{dx}\!\left[\frac{4}{x^2} - 6\sqrt{x} + 9\right] \).

1
Rewrite: \( 4x^{-2} - 6x^{1/2} + 9 \)
2
Differentiate term-by-term: \( 4(-2)x^{-3} - 6 \cdot \frac{1}{2}x^{-1/2} + 0 \)
3
Simplify: \( -8x^{-3} - 3x^{-1/2} \)
4
Optional rewrite: \( -\frac{8}{x^3} - \frac{3}{\sqrt{x}} \)

Example 3: Multi-term Expression

Let \( g(x) = \frac{1}{2}x^6 - 3x^4 + 8x^3 - x + 10 \). Find \( g'(x) \).

$$ g'(x) = \frac{1}{2} \cdot 6x^5 - 3 \cdot 4x^3 + 8 \cdot 3x^2 - 1 + 0 $$

$$ g'(x) = 3x^5 - 12x^3 + 24x^2 - 1 $$

Graphical Interpretation

x y -2 -1 1 2 1 -1
\( f(x) = 0.25x^3 - 1.5x \)
\( f'(x) = 0.75x^2 - 1.5 \)

The original function \( f(x) \) and its derivative \( f'(x) \). Where \( f'(x) = 0 \), the original function has a horizontal tangent (local max or min).

Common Mistakes

✕ Forgetting the Constant
Writing \( \frac{d}{dx}[7] = 7 \) instead of \( 0 \). Constants always differentiate to zero.
✕ Dropping the Coefficient
Writing \( \frac{d}{dx}[5x^3] = 3x^2 \) instead of \( 15x^2 \). Don't forget to multiply by the constant.
✕ Wrong Rewrite
Treating \( \frac{1}{x^2} \) as \( x^2 \) instead of \( x^{-2} \). Rewrite before differentiating.

Question 1 of 5

What is \( \frac{d}{dx}[42] \)?

Question 2 of 5

Find \( \frac{d}{dx}[4x^5 + 3x^2] \).

Question 3 of 5

If \( h(x) = 7x^3 - 2x + 5 \), what is \( h'(x) \)?

Question 4 of 5

What is \( \frac{d}{dx}\!\left[\frac{6}{x^3}\right] \)?

Question 5 of 5

Let \( f(x) = 2x^4 - 5x^3 + x^2 - 8x + 3 \). What is \( f'(1) \)?