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

Facts & Assumptions

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

[L1]

$a^{x} = \exp (x \log a)$ and $x^{α} = \exp (α \log x)$ (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: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , A function differentiable at $c$ is continuous at $c$ ).

[L4]

Positive-base powers satisfy $x^{r + s} = x^{r} x^{s}$ (The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents); together with [L1] and $\exp (\log x) = x$ , this gives $x^{α - 1} = x^{α} / x$ .

Proof

technique · direct

1.1L1L2L3

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

1.2L1L2L3L4

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

2.1step 1.1step 1.2L3∎

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

Continuity and derivatives of positive-base real powers

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources