Proofs

Browse a catalog of interesting and common proofs, from core analysis and linear algebra arguments to standard proof techniques worth knowing well.

Showing 4 proofs in Calculus.

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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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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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Proof of the chain rule

The chain rule says the derivative of $f (g (x))$ is $f^{'} (g (x)) \cdot g^{'} (x)$ . The proof uses a continuous extension of the difference quotient to handle all cases cleanly.

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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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Proofs

Proof that the derivative of a constant function is zero

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

Proof of the chain rule

Proof that a differentiable function is continuous

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