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The inverse $E$ is differentiable, $E^{'} = E$ , and $E (0) = 1$

Statement

The inverse function $E : R \to (0, \infty)$ is differentiable and

$E^{'} (y) = E (y) (y \in R), E (0) = 1.$

Facts & Assumptions

Given: $y \in R$ .

[F1]

$E = L^{- 1}$ , $L (E (y)) = y$ , and $E (L (x)) = x$ (The integral exponential $E : R \to (0, \infty)$ as the inverse of $L$ ).

[L1]

$L^{'} (x) = 1 / x$ for $x > 0$ , and $L (1) = 0$ (The integral logarithm satisfies $L^{'} (x) = 1 / x$ for $x > 0$ and $L (1) = 0$ ).

[L2]

If a continuous injective function on a nondegenerate interval has a nonzero derivative at $c$ , then its inverse is differentiable at $f (c)$ with derivative $1 / f^{'} (c)$ (Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f^{'} (c) \neq 0$ , then the inverse $g$ is differentiable at $f (c)$ with $g^{'} (f (c)) = 1 / f^{'} (c)$ ; and if $f^{'} (c) = 0$ then $g$ is not differentiable at $f (c)$ ).

[L3]

A differentiable function is continuous (A function differentiable at $c$ is continuous at $c$ ).

Proof

technique · direct

1.1

The function $L$ is injective because it has inverse $E$ , and it is continuous by [L1] and [L3]. At $c = E (y) > 0$ its derivative is $L^{'} (c) = 1 / c \neq 0$ .

F1L1L3

1.2

Since $L (1) = 0$ , the inverse identity gives $E (0) = 1$ .

L1F1

2.1

Apply [L2] at $c = E (y)$ . Since $L (c) = y$ , $E^{'} (y) = \frac{1}{L^{'} (E (y))} = E (y) .$

step 1.1L2L1F1algebra

3.1

The arbitrary choice of $y$ , together with steps 2.1 and 1.2, proves all claims.

step 2.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 67 results over 20 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

OpenStax, Calculus Volume 1, Section 6.7 (standard reference, not scraped)

The inverse EE is differentiable, E′=EE′=E, and E(0)=1E(0)=1