Proof that the derivative of a constant function is zero

Primary Tool

Proof Builder

This proof is easy to compress into one sentence, but the proof-builder layout makes the underlying logic explicit: write the quotient, substitute the constant values, simplify the numerator, and then take the limit.

Why the quotient collapses immediately

If $f (x) = c$ , then both $f (x + h)$ and $f (x)$ are the same number $c$ . That means the numerator in the difference quotient is $c - c = 0$ before you do any serious limit manipulation.

So the derivative is zero not because we declare it to be, but because the derivative definition itself reduces to the limit of the constant expression $0$ .

What this proof is really teaching

This page is useful because it separates two ideas students often blend together: a constant function has a flat graph, and the derivative definition proves that flatness algebraically.

The theorem is small, but it sets a standard for later derivative proofs: write the quotient first, simplify honestly, and only then take the limit.

It also explains why constant terms disappear later inside larger derivative computations. When a rule produces the derivative of a constant, this proof is the reason that contribution becomes zero.

Common Pitfall

The derivative is not zero just because the graph 'looks flat.' The proof works because the difference quotient itself becomes $0$ for every nonzero $h$ .

Try a Variation

Use the same method on a linear function $f (x) = m x + b$ . Which part stays the same, and which term survives after simplification?

Keep moving through the cluster

Back to Learn Math →

Worked ExampleCalculusMath Workspace (CAS)

Differentiate a polynomial from first principles

This worked example shows how the derivative definition creates an algebra problem first and a limit problem second. The simplification step is the point of the exercise.

Open worked example →

ProofCalculusProof Builder

Proof that $\frac{d}{d x} x^{n} = n x^{n - 1}$ for positive integers

This proof shows where the power rule comes from rather than treating it as a black box. The binomial theorem isolates the only term that survives after dividing by $h$ and taking the limit.

Open proof →

ExplanationCalculus2D Graphing

What a derivative means geometrically

The derivative is the slope of the tangent line, but that phrase only becomes useful when you compare the curve to nearby secant lines and local linear approximations.

Open explanation →