Q.When is a function $f(x)$ said to be differentiable at $x=a$?

The derivative grew out of two of the four classic 17th-century calculus problems: the tangent line problem and the velocity problem. Both reduce to the same limiting construction.

From secant to tangent. For a curve $y=f(x)$ and a point $P(x_0,f(x_0))$, a second point $Q(x_0+\Delta x,\,f(x_0+\Delta x))$ determines a secant line with slope (the difference quotient)

Letting $Q\to P$ (i.e. $\Delta x\to0$) makes the secant line approach the tangent line, whose slope is

provided this limit exists (Definition 10.1). Since a point and a slope fix a unique line, the tangent line — when it exists — is unique.

From average to instantaneous velocity. If $s=f(t)$ is the position of an object moving on a line, the average velocity over $[t_0,t_0+\Delta t]$ is $v_{\text{avg}}=\dfrac{f(t_0+\Delta t)-f(t_0)}{\Delta t}$ — again a secant slope, now on the position–time graph. Shrinking $\Delta t\to0$ gives the instantaneous velocity

The derivative, in general. Both constructions are the same limit applied to different functions, so calculus gives it one name:

Definition 10.2. $f$ is differentiable at $x_0$ if $f'(x_0)=\displaystyle\lim_{\Delta x\to0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$ exists. For every $x$ where the limit exists, $f'(x)$ defines the derivative function, and finding it is called differentiation.

Notations. $f'(x)$, $y'$, $\dfrac{dy}{dx}$, $\dfrac{d}{dx}f(x)$, $Dy$ all denote the same object. $\dfrac{dy}{dx}$ is read "derivative of $y$ w.r.t. $x$" — it is a single symbol for a limit, not literally a fraction $dy\div dx$, even though later techniques (implicit and parametric differentiation) manipulate it formally as one. …