Differentiability

A function $f$ is differentiable at $a$ iff:

\lim_{x\to a}{\frac{f(x)-f(a)}{x-a}} = L \in \mathbb{R} = f'(a)

When it is differentiable, $f'(a)$ is called the derivative of $f$ at $a$ .

$c\in[a,b]$ is called a critical point iff:

f\text{ is not differentiable at } c \;\; \lor \;\; f'(c)=0

One-Side Differentiable

$f$ is differentiable at $a$ iff $f$ is left differentiable at $a$ and $f$ is right differentiable at $a$ .

A function $f$ is left-differentiable at $a$ iff:

\lim_{x\to a^{-}}{\frac{f(x)-f(a)}{x-a}} = L \in \mathbb{R} = f'_{-}(a)

A function $f$ is right-differentiable at $a$ iff:

\lim_{x\to a^{+}}{\frac{f(x)-f(a)}{x-a}} = L \in \mathbb{R} = f'_{+}(a)

A function $f$ is differentiable in $(a,b)$ iff $f$ is differentiable on every $c\in(a,b)$ .

A function $f$ is differentiable in $[a,b]$ iff $f$ is:

A function $f$ is said to be continously differentiable at $a$ iff :

$f \text{ is differentiable at } a \implies f \text{ is continuous at } a$

Likewise, one-sided differentiability implies corresponding one-sided continuity.

\frac{\text{d}}{\text{d}x} (f \pm g) = f' \pm g'

\frac{\text{d}}{\text{d}x} (fg) = fg' + fg'

\frac{\text{d}}{\text{d}x} \bigg(\frac{f}{g}\bigg) = \frac{gf' - fg'}{g^2}

\frac{\text{d}}{\text{d}x} f(g(x)) = f'(g(x))\,g'(x)

\frac{\text{d}}{\text{d}x} f^n = nf^{n-1}(x) f'(x)

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October 17, 2026 1 min read

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