Exponential and logarithmic derivatives

Fact: $B (x) = b^{x}$ is continuous.

Fact: $B^{'} (0) = lim_{ℎ \to 0} \frac{b^{0 + ℎ} - b^{0}}{ℎ}$ exists.

Fact: there is a unique value $𝒆$ such that

E^{'} (0) = {(𝒆^{x})}^{'} |_{x = 0} = 1

If $ℬ (x) = 𝒷^{x}$ then

ℬ^{'} (x) = ℬ^{'} (0) 𝒷^{x}

If $ℰ (x) = ℯ^{x}$ then

ℰ^{'} (g (x)) = ℯ^{g (x)} g^{'} (x)

If $x > 0$ and $y = ln x$ then

y^{'} = {[ln x]}^{'} = \frac{1}{x}

If $g (x) > 0$ and $h (x) = ln (g (x))$ then

h^{'} (x) = \frac{1}{g (x)} g^{'} (x)

Let $b > 0, b \neq 1$

Examples

Compute derivatives of

Compute derivatives of

Compute derivatives of

Compute derivatives