Table of Contents
The Derivative as a Limit
In the parent chapter on derivatives, the basic idea of a derivative as an “instantaneous rate of change” or “slope of a tangent line” has already been introduced. Here we focus on making that idea precise with a formal, limit-based definition, and on a few closely related definitions and notations.
We will not yet develop rules for computing derivatives efficiently (that belongs to “Differentiation rules”), and we will not apply derivatives to problems (that belongs to later chapters). Here the emphasis is: what the derivative actually is, as a mathematical object.
The Basic Limit Definition
Consider a real-valued function $f$ defined on some interval of real numbers, and a point $a$ in its domain.
We look at the difference quotient:
\[
\frac{f(a+h)-f(a)}{h},
\]
where $h$ is a small nonzero number. Geometrically, this fraction is the slope of the secant line through the points $(a,f(a))$ and $(a+h,f(a+h))$.
If, as $h$ gets closer and closer to $0$, these slopes approach a single number, then that number is the derivative of $f$ at $a$.
Formally, we define:
\[
f'(a) = \lim_{h \to 0}\frac{f(a+h)-f(a)}{h},
\]
provided this limit exists (and is finite).
This is the definition of the derivative at a point.
Alternative but Equivalent Form: $x \to a$
Sometimes, instead of introducing the small increment $h$, we write the derivative in terms of $x$ approaching $a$:
\[
f'(a) = \lim_{x \to a}\frac{f(x)-f(a)}{x-a},
\]
again provided the limit exists.
These two forms are equivalent:
- In the $h$-form, we think: “Let $h$ be the horizontal change from $a$ to $a+h$.”
- In the $x$-form, we think: “Let $x$ be the variable point approaching $a$.”
The substitution $x = a + h$ transforms one form into the other.
Notation for the Derivative
Several standard notations are used for derivatives. If $y = f(x)$, then the derivative of $f$ with respect to $x$ can be written as:
- $f'(x)$ (read: “$f$ prime of $x$”),
- $\dfrac{dy}{dx}$ (Leibniz notation, read: “$dy$ by $dx$”),
- $\dfrac{d}{dx} f(x)$ (an operator view: “the derivative with respect to $x$ of $f(x)$”),
- $y'$ (especially when $y$ is given as a function of $x$).
At a specific point $x=a$, we might write:
- $f'(a)$,
- $\left.\dfrac{dy}{dx}\right|_{x=a}$,
- $y'(a)$.
These all represent the same number, defined by the limit above.
Interpreting the Difference Quotient
The difference quotient
\[
\frac{f(a+h)-f(a)}{h}
\]
has both a numerical and a geometric meaning.
- Numerical meaning (rate of change): It measures the average rate of change of $f$ between $x=a$ and $x=a+h$.
- Geometric meaning (slope): It is the slope of the secant line through $(a,f(a))$ and $(a+h,f(a+h))$.
As $h \to 0$, if those average rates (or slopes of secant lines) settle down to a single value, that limiting value is the instantaneous rate of change at $x=a$, i.e., the slope of the tangent line at $(a,f(a))$. This “settling down” is exactly what the limit in the definition captures.
When the Derivative Exists
By definition, $f$ is differentiable at $a$ if the limit
\[
\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}
\]
exists and is a finite real number.
If this limit exists at every point in some interval, we say $f$ is differentiable on that interval.
Not every function is differentiable everywhere. Situations where the derivative fails to exist at $a$ include (we only describe them briefly here, without analysis):
- A corner or cusp at $x=a$ (the left and right “slopes” do not agree).
- A vertical tangent, where the slopes grow without bound.
- A discontinuity at $x=a$.
The precise relationships between differentiability and continuity are handled in the chapter on continuity; here the key point is simply that the derivative is a limit, and limits do not always exist.
One-Sided Derivatives
Sometimes it is useful to look at derivatives from just one side. We can define:
- The right-hand derivative at $a$:
\[
f'_+(a) = \lim_{h \to 0^+}\frac{f(a+h)-f(a)}{h},
\] - The left-hand derivative at $a$:
\[
f'_-(a) = \lim_{h \to 0^-}\frac{f(a+h)-f(a)}{h}.
\]
If both one-sided derivatives exist and are equal, then the (two-sided) derivative exists and equals that common value:
\[
f'(a) \text{ exists } \iff f'_+(a) \text{ and } f'_-(a) \text{ both exist and } f'_+(a)=f'_-(a).
\]
One-sided derivatives are especially relevant for functions defined only on one side of a point, or for understanding sharp corners.
Derivative as a Function
So far we have focused on the derivative at a point $a$. Often we are interested in how the derivative varies as $x$ varies.
If $f$ is differentiable at every point in some interval, we can define a new function $f'$ by:
\[
f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h},
\]
for each $x$ in the interval.
This function $f'$ is called the derivative of $f$, or the first derivative of $f$.
Thus, the derivative is itself a function, not just a single number. Later chapters will study properties of this derivative function and use it to analyze and sketch graphs, solve optimization problems, and so on.
Higher-Order Derivatives (Brief Introduction)
The limit definition above gives the first derivative. If $f'(x)$ is itself differentiable, we can differentiate again to obtain the second derivative:
\[
f''(x) = \frac{d}{dx}\big(f'(x)\big) = \frac{d^2 y}{dx^2}.
\]
More generally, if $f$ is differentiable many times, we can define the $n$-th derivative $f^{(n)}(x)$ for positive integers $n$, by repeatedly applying the derivative operation. The detailed uses of higher-order derivatives belong in later chapters; here it is enough to note that they are obtained by repeatedly using the same basic limit process.
Summary of the Definition
To summarize the essential content of this chapter:
- The derivative of $f$ at $a$ is defined by
\[
f'(a) = \lim_{h \to 0}\frac{f(a+h)-f(a)}{h},
\]
if this limit exists and is finite. - Equivalently,
\[
f'(a) = \lim_{x \to a}\frac{f(x)-f(a)}{x-a}.
\] - We use notations such as $f'(x)$, $y'$, $\dfrac{dy}{dx}$, and $\dfrac{d}{dx}f(x)$.
- If $f'(x)$ exists for each $x$ in some interval, the derivative defines a new function $f'$ on that interval.
All computational techniques and applications of derivatives build on this foundational limit-based definition.