Explicit Differentiation／Derivative／Differentiable／Differential Functions (differentiable at a point - differentiable over an interval - differentiable everywhere／over the entire domain)

measures the instantaneous rate of change of the output variable with respect to the input variable
a function is differentiable at point 𝑝 if the derivative exists at 𝑝, 𝑓’(𝑝), i.e. the following two limits must be equal to each other:
- $lim_{𝛿 \to 0^{+}} \frac{f ( p + 𝛿 ) - f ( p )}{𝛿}$
- $lim_{𝛿 \to 0^{-}} \frac{f ( p ) - f ( p - 𝛿 )}{𝛿}$
- $Thus f^{'} (p) = lim_{𝛿 \to 0^{+}} \frac{f ( p + 𝛿 ) - f ( p )}{𝛿} = lim_{𝛿 \to 0^{-}} \frac{f ( p ) - f ( p - 𝛿 )}{𝛿}$
a function is differentiable over an interval if it is differentiable at every point within the interval
a function is differentiable (everywhere or over the entire domain) if it is differentiable at every point within the domain
a function that is NOT differentiable is a non-differentiable function
a function that is differentiable (at a point | over an interval | everywhere) implies the function is continuous respectively
opposite: integral calculus

／var／log marcus chiu