Definition of Derivative

Formal definition. Let $f$ be a function defined in some open interval

containing the point $x=a$. The derivative of $f$ at $a$, written $f'(a)$, is defined as

provided this limit exists. An entirely equivalent form, obtained by writing $x=a+h$ (so

$h=x-a$, and $h\to0$ exactly when $x\to a$), is

Both forms are used interchangeably; the first is usually more convenient for algebraic

manipulation (expanding $f(a+h)$), while the second matches the "slope between two points"

picture of Section 7 most directly.

Differentiability. If the limit above exists, $f$ is said to be differentiable at $a$.

Finding $f'(x)$ for a general point $x$ directly from this limit -- rather than by quoting an

already-known formula -- is called differentiating from first principles (also called ab initio differentiation), the technique this section's exercises practise directly.

One-sided derivatives. Exactly as with limits (Section 1), a left-hand derivative

and a right-hand derivative

can be defined using one-sided limits. The full (two-sided) derivative $f'(a)$ exists if and

only if both one-sided derivatives exist and are equal; a function whose graph has a sharp

"corner" at $a$ -- such as $f(x)=|x|$ at $x=0$ -- typically has unequal one-sided derivatives

there, so $f$ fails to be differentiable at that point even though $f$ itself is perfectly

well-defined and continuous there.

Notation. Several equivalent notations for the derivative are used throughout mathematics

and appear across this chapter and beyond: $f'(x)$ (Lagrange's notation), $\dfrac{dy}{dx}$ or

$\dfrac{d}{dx}\big[f(x)\big]$ (Leibniz's notation, writing $y=f(x)$), and occasionally $Df(x)$.

All three denote exactly the same quantity.

Worked illustration. For $f(x)=x^2+3x$, …