TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Continuity and derivatives of positive-base real powers

Statement

For $a>0$ , the function $x\mapsto a^x$ is continuous on $\mathbb R$ and $(a^x)'=a^x\log a.$ For $\alpha\in\mathbb R$ , the function $x\mapsto x^\alpha$ is continuous and differentiable on $(0,\infty)$ , with $(x^\alpha)'=\alpha x^{\alpha-1}.$

Facts & Assumptions

Given: A positive base $a$ , a real exponent $\alpha$ , and $x>0$ .

[L1]

$a^x=\exp(x\log a)$ and $x^\alpha=\exp(\alpha\log x)$ (Real powers for positive bases, with the zero-base positive-exponent convention).

[L2]

$\log'(x)=1/x$ on $(0,\infty)$ and $\exp'(u)=\exp(u)$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, The exponential function is smooth and $(\exp)'=\exp$ ).

[L3]

The chain rule and algebra of derivatives apply to differentiable real functions, and differentiability implies continuity (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$ , Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$ , $(\alpha f)'(c) = \alpha f'(c)$ , $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$ , and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$ , A function differentiable at $c$ is continuous at $c$ ).

Proof

technique · direct

1.1

The chain rule applied to $a^x=\exp(x\log a)$ gives $(a^x)'=\exp(x\log a)\log a=a^x\log a$ .

L1L2L3

1.2

The chain rule applied to $x^\alpha=\exp(\alpha\log x)$ gives $(x^\alpha)'=\exp(\alpha\log x)\alpha/x=\alpha x^{\alpha-1}$ .

L1L2L3

2.1

Both functions are continuous on their stated domains because the displayed derivatives exist there.

step 1.1step 1.2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 97 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)