Differentiate a polynomial from first principles

Primary Tool

Math Workspace (CAS)

CAS is useful here because it keeps the algebraic expansion, the simplification, and the limiting step in one transcript.

Problem

Find the derivative of $f (x) = x^{2} + 3 x$ from the definition $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ .

This is a standard exercise because it shows that the derivative formula comes from algebraic cancellation, not from a rule you memorize first.

What to watch

When you substitute $x + h$ , the numerator looks messy on purpose. The cancellation is what reveals the derivative.

The limit is easy only after the numerator is simplified to $2 x + h + 3$ . Sending $h$ to $0$ too early loses the structure of the problem.

Step-by-step walkthrough

The goal is to make the limit readable by turning the difference quotient into a simpler algebraic expression.

1Start from $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ and substitute $f (x) = x^{2} + 3 x$ .
2Expand $f (x + h)$ as $(x + h)^{2} + 3 (x + h) = x^{2} + 2 x h + h^{2} + 3 x + 3 h$ .
3Subtract $f (x) = x^{2} + 3 x$ so the numerator becomes $2 x h + h^{2} + 3 h$ .
4Factor out $h$ to get $\frac{h ( 2 x + h + 3 )}{h}$ , then cancel the common factor for $h \neq = 0$ .
5Take the limit of the simplified expression $2 x + h + 3$ as $h \to 0$ to obtain $2 x + 3$ .

Common Pitfall

Do not substitute $h = 0$ into the original quotient. First simplify the expression, then take the limit of the simplified form as $h$ approaches $0$ .

Try a Variation

Try the same method on $f (x) = x^{3}$ . Which cancellation pattern repeats, and where does the extra power show up?

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ProofCalculusProof Builder

Proof that the derivative of a constant function is zero

This proof is short, but it matters because it shows the derivative definition already knows that a function with no change has zero slope. Nothing extra has to be added as a rule.

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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.

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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.

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ProofCalculusProof Builder

Proof that a differentiable function is continuous

This proof shows why differentiability is a stronger condition than continuity. The difference $f (a + h) - f (a)$ factors into a difference quotient times $h$ , and that product is forced to zero.

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Differentiate a polynomial from first principles

Problem

What to watch

Step-by-step walkthrough

Common Pitfall

Try a Variation

Keep moving through the cluster

Proof that the derivative of a constant function is zero

Proof that dxd​xn=nxn−1 for positive integers

What a derivative means geometrically

Proof that a differentiable function is continuous

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