Definition of Derivative and Differentiability

Definition. Let $f(x)$ be a function defined on an open interval containing the point $a$. If

exists, then $f$ is said to be differentiable at $x=a$, and this limit value is called the derivative of $f$ at $a$, written $f'(a)$.

The same idea works at any point $x$ in the domain of $f$, not just at one fixed point $a$. Let $y=f(x)$. Give $x$ a small increment $\delta x$; correspondingly $y$ picks up a small increment $\delta y$, so that

Since $y=f(x)$, subtracting gives

Because $\delta x$ is a genuine (nonzero) increment, both sides can be divided by $\delta x$:

Now let $\delta x\to0$:

If this limit exists, it is called the derivative of the function, written $\dfrac{dy}{dx}$, so that

Equivalently, thinking of the graph of $f$, i.e. the set of points $\{(x,y): y=f(x)\}$, the same limit can be written purely in terms of $x$ and $y$: if $y=f(x)$ is differentiable then

Left-hand and right-hand derivatives. Writing $\delta x=h$, suppose $\displaystyle\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$ exists. That is the same as saying its left-hand limit and right-hand limit both exist and are equal:

$\lim_{h\to0^-}\dfrac{f(a+h)-f(a)}{h}=\lim_{h\to0^+}\dfrac{f(a+h)-f(a)}{h}$ …