Derivative of Constants and the Identity
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 3.3: “Differentiation Rules” |
| Book URL | https://openstax.org/details/books/calculus-volume-1 |
Freely available and openly licensed.
Key idea
Two of the most basic differentiation facts follow directly from the definition with almost no algebra:
- A constant function $f(x) = c$ never changes as $x$ changes, so its rate of change is zero.
- The identity function $f(x) = x$ changes at exactly the same rate as $x$, so its rate of change is 1.
These are the seeds of every differentiation rule that follows. Every polynomial rule, for instance, is built from these two facts combined with the sum rule, constant multiple rule, and power rule.
Prerequisite Check
Quick Reference
\[ \frac{d}{dx}[c] = 0 \qquad \frac{d}{dx}[x] = 1 \]
Key Concepts
1. Derivative of a Constant Function
Theorem. If $f(x) = c$ (a constant), then $f'(x) = 0$.
Proof from definition. \[ f'(x) = \lim_{h \to 0} \frac{c - c}{h} = \lim_{h \to 0} \frac{0}{h} = \lim_{h \to 0} 0 = 0. \]
Geometric meaning. The graph of $y = c$ is a horizontal line. Its slope is zero everywhere. The tangent at every point is the line itself.
Examples:
| Function | Derivative |
|---|---|
| $f(x) = 7$ | $f'(x) = 0$ |
| $f(x) = -\pi$ | $f'(x) = 0$ |
| $f(x) = \sqrt{2}$ | $f'(x) = 0$ |
2. Derivative of the Identity
Theorem. If $f(x) = x$, then $f'(x) = 1$.
Proof from definition. \[ f'(x) = \lim_{h \to 0} \frac{(x+h) - x}{h} = \lim_{h \to 0} \frac{h}{h} = \lim_{h \to 0} 1 = 1. \]
Geometric meaning. The graph of $y = x$ has slope 1 everywhere. The tangent at every point is the line itself, with slope 1.
3. Applying Both Rules Together
These facts combine with the sum rule and constant multiple rule to differentiate any linear function.
Example 1. If $f(x) = 5x - 3$, find $f'(x)$.
$\dfrac{d}{dx}[5x - 3] = 5 \cdot \dfrac{d}{dx}[x] - \dfrac{d}{dx}[3] = 5(1) - 0 = 5$.
The derivative of a linear function is its slope. The constant $-3$ shifts the graph up and down but does not affect the rate of change.
Example 2. If $g(x) = -2x + 11$, find $g'(x)$.
$g'(x) = -2(1) + 0 = -2$.
4. Why Derivatives of Constants Vanish
Intuitively: if a quantity never changes, its rate of change is zero. Adding a constant to a function shifts its graph vertically but does not tilt it. Therefore adding a constant to $f(x)$ does not change $f'(x)$.
Consequence. For any differentiable $f$: $\dfrac{d}{dx}[f(x) + c] = f'(x)$. Constants “evaporate” under differentiation.
Common Errors
| Error | Example | Correction |
|---|---|---|
| Derivative of a constant is the constant | “$\frac{d}{dx}[7] = 7$” | Derivative is 0; a constant has zero rate of change |
| Derivative of $x$ is 0 | Confusing $f(x) = 1$ (constant) with $f(x) = x$ | $\frac{d}{dx}[x] = 1$; $\frac{d}{dx}[1] = 0$ |
Leveled Practice
Level 1 -- Direct Application
Problem 1. Differentiate each: (a) $f(x) = 100$, (b) $g(x) = x$, (c) $h(x) = e$ (Euler’s number).
Show answer
(a) $f'(x) = 0$. (b) $g'(x) = 1$. (c) $h'(x) = 0$ ($e$ is a constant).
Problem 2. Find the derivative of $y = 3x - 8$.
Show answer
$\frac{d}{dx}[3x - 8] = 3(1) - 0 = 3$.
Level 2 -- Understanding the Rule
Problem 3. Let $f(x) = x + 5$ and $g(x) = x$. Compare $f'(x)$ and $g'(x)$.
Show answer
$f'(x) = 1$ and $g'(x) = 1$. The constant $5$ shifts the graph of $g$ up by 5 but does not change its slope. Both functions have the same derivative everywhere.
Level 3 -- Connecting to Geometry
Problem 4. At what points does the tangent line to $y = x + 4$ have slope equal to 1? Describe the geometric meaning.
Show answer
$\frac{d}{dx}[x + 4] = 1$ everywhere. The tangent has slope 1 at every point. Geometrically, $y = x + 4$ is itself a line, and every tangent to a line is the line itself -- slope 1 at all points.
Common Misconceptions
the derivative of a constant is the constant itself. A constant function $f(x) = 7$ has a perfectly flat graph: the slope is zero everywhere. The derivative is $f'(x) = 0$, not 7. The value 7 describes the height of the graph, not how fast it is changing. A graph that never rises or falls has zero rate of change at every point.
$\frac{d}{dx}[x] = x$ because “the derivative of a variable is itself.” The derivative of $f(x) = x$ is $f'(x) = 1$, the slope of the line $y = x$, which is 1 everywhere. Students sometimes confuse differentiation with the identity function. The confusion often arises from applying the power rule incorrectly: $\frac{d}{dx}[x^1] = 1 \cdot x^{1-1} = 1 \cdot x^0 = 1$.
Mastery Checklist
Mental Model
Constants are invisible to derivatives. A constant $c$ adds the same amount to $f(x)$ for every $x$ -- so the difference $f(x+h) - f(x)$ cancels the constant out completely, and the rate of change is zero.
The identity $f(x) = x$ changes by exactly $h$ when $x$ changes by $h$, so the ratio is always $h/h = 1$, and the rate of change is exactly 1.
Connections
Within MATH161
- Power rule: $\frac{d}{dx}[c] = 0$ is the power rule at $n = 0$; $\frac{d}{dx}[x] = 1$ is the power rule at $n = 1$.
- Integration: The antiderivative of $0$ is a constant $+C$, explaining why indefinite integrals always include $+C$.