Definition of the Derivative

A Pointwise Definition and a Function of Derivatives

In “The Derivative as a Limit,” differentiability at an interior point was introduced through a limit of difference quotients. Here we make that definition precise as a pointwise property, then express it in a form that isolates the derivative’s role: it is the coefficient of the linear change in the function, up to an error that becomes negligible compared with the input change.

Throughout this tutorial, let $I$ be an interval, let $f:I\to\mathbb{R}$, and let $a$ be an interior point of $I$. Because $a$ is interior, $a+h$ remains in $I$ for all sufficiently small $h$. The increment $h$ is nonzero when it appears in a difference quotient.

Definition: The function $f$ is differentiable at $a$ if the finite limit $$ \lim_{h\to0}\frac{f(a+h)-f(a)}{h} $$ exists. Its value is the derivative of $f$ at $a$, written $f'(a)$.

This definition assigns a number to a particular point $a$, provided the limit exists. It does not assert that the limit exists at every point of $I$. The set of points where it does exist can itself be used as the domain of a new function.

Definition: The derivative function of $f$ is the function $f'$ whose domain is $$ D=\{a\in I: f\text{ is differentiable at }a\} $$ and whose value at each $a\in D$ is the derivative $f'(a)$ defined by the difference-quotient limit.

If $D=I$, then $f'$ is defined throughout $I$. If differentiability fails at even one point, the derivative function is not defined there. In particular, writing $f'(a)$ is meaningful only after differentiability at $a$ has been established.

Worked Examples: Applying the Definition

Worked Example: A Quadratic at a Fixed Point

Let $f(x)=x^2+3x$, and find its derivative at $a=1$ directly from the definition. First, $f(1)=4$. For $h\ne0$,

\frac{f(1+h)-f(1)}{h} =\frac{(1+h)^2+3(1+h)-4}{h}.

Expanding gives $(1+h)^2+3(1+h)-4=1+2h+h^2+3+3h-4=5h+h^2$. Thus, for nonzero $h$,

\frac{f(1+h)-f(1)}{h} =\frac{5h+h^2}{h} =5+h.

As $h\to0$, $5+h\to5$. Therefore $f$ is differentiable at $1$, and $f'(1)=5$. The cancellation uses $h\ne0$; the limit then concerns the simplified expression as nonzero $h$ approaches zero.

Worked Example: Finding a Derivative Function

Let $g(x)=x^4$ on $\mathbb{R}$. Fix any $a\in\mathbb{R}$. For $h\ne0$, expanding $(a+h)^4$ gives

\frac{g(a+h)-g(a)}{h} =\frac{(a+h)^4-a^4}{h} =4a^3+6a^2h+4ah^2+h^3.

For each fixed $a$, the terms $6a^2h$, $4ah^2$, and $h^3$ tend to zero as $h\to0$. Hence the limit exists and equals $4a^3$. Since $a$ was arbitrary, $g$ is differentiable at every real number, and its derivative function is $g'(x)=4x^3$ for every $x\in\mathbb{R}$.

Worked Example: Different Remainders on the Two Sides

Define $u:\mathbb{R}\to\mathbb{R}$ by $u(x)=x+x^2$ when $x\geq0$ and $u(x)=x-x^2$ when $x<0$. At $a=0$, $u(0)=0$. If $h>0$, then $u(h)=h+h^2$, so the quotient is $1+h$. If $h<0$, then $u(h)=h-h^2$, so the quotient is $1-h$. In both cases, the quotient tends to $1$ as $h\to0$.

Therefore $u$ is differentiable at $0$, with $u'(0)=1$. The expressions on the two sides differ, but each differs from $1$ by a term that tends to zero. Differentiability requires the two-sided limit to exist; it does not require the function to be given by the same formula on both sides.

Differentiability as a First-Order Approximation

The difference quotient definition can be rearranged to show what the derivative says about the function values near $a$. When the derivative exists, the increment in the function can be written as a linear term $f'(a)h$ plus a remainder. The crucial condition is that the remainder divided by $h$ tends to zero.

Definition: A remainder $r(h)$, defined for sufficiently small $h$, is negligible compared with $h$ as $h\to0$ if $r(0)=0$ and $$ \lim_{h\to0,\ h\ne0}\frac{r(h)}{h}=0. $$

The condition is stronger than merely requiring $r(h)\to0$. For instance, a remainder might tend to zero at the same rate as $h$, in which case $r(h)/h$ need not tend to zero. What differentiability requires is that, relative to the size of the input increment, the remainder becomes insignificant.

Theorem (First-Order Approximation Characterization): Let $a$ be an interior point of $I$, and let $L\in\mathbb{R}$. The function $f$ is differentiable at $a$ with $f'(a)=L$ if and only if there is a remainder $r(h)$, defined for all sufficiently small $h$, such that $$ f(a+h)=f(a)+Lh+r(h) $$ and $$ \lim_{h\to0,\ h\ne0}\frac{r(h)}{h}=0. $$

Proof. Suppose first that $f$ is differentiable at $a$ and $f'(a)=L$. Define $r(0)=0$, and for small nonzero $h$ define

$$ r(h)=f(a+h)-f(a)-Lh. $$

By this definition, $f(a+h)=f(a)+Lh+r(h)$. For nonzero $h$, division by $h$ gives

\frac{r(h)}{h} =\frac{f(a+h)-f(a)}{h}-L.

The difference quotient tends to $L$ by differentiability, so the right-hand side tends to zero. Thus $r(h)/h\to0$, as required.

Conversely, suppose such a remainder exists. For nonzero $h$, rearranging the stated approximation and dividing by $h$ gives

\frac{f(a+h)-f(a)}{h} =L+\frac{r(h)}{h}.

By assumption, $r(h)/h\to0$. Therefore the difference quotient tends to $L$. By the definition of the derivative, $f$ is differentiable at $a$ and $f'(a)=L$. This proves both directions. $\square$

This characterization is often written informally as $f(a+h)=f(a)+Lh+o(h)$, where $o(h)$ denotes a remainder whose ratio to $h$ tends to zero. The explicit limit is the important part: it states exactly how small the error must be relative to the increment.

The Linear Coefficient Is Unique

The approximation characterization also shows that there cannot be two different derivative values at the same point. The derivative is not merely one possible coefficient that gives a reasonable approximation; it is the only coefficient whose error is negligible compared with $h$.

Theorem (Uniqueness of the First-Order Coefficient): Suppose $a$ is an interior point of $I$, and suppose $L,M\in\mathbb{R}$ both satisfy first-order approximations $$ f(a+h)=f(a)+Lh+r(h) $$ and $$ f(a+h)=f(a)+Mh+s(h), $$ where $r(h)/h\to0$ and $s(h)/h\to0$ as $h\to0$, $h\ne0$. Then $L=M$.

Proof. For every sufficiently small nonzero $h$, equate the two expressions for $f(a+h)$ and subtract $f(a)$. This gives $Lh+r(h)=Mh+s(h)$, and hence

L-M=\frac{s(h)}{h}-\frac{r(h)}{h}.

As $h\to0$, both terms on the right tend to zero, so the right-hand side tends to zero. The left-hand side is the fixed number $L-M$, independent of $h$. Therefore $L-M=0$, and $L=M$. $\square$

Together, the two theorems establish the exact relationship between the limit definition and the approximation viewpoint: a derivative exists precisely when there is a first-order linear approximation with a negligible remainder, and the coefficient of that approximation is uniquely determined.

Reading and Using the Definition Carefully

For the piecewise function $u$ above, the first-order approximation at zero is especially transparent. When $h>0$, $u(h)=u(0)+1\cdot h+h^2$; when $h<0$, $u(h)=u(0)+1\cdot h-h^2$. In either case, the remainder divided by $h$ tends to zero: it is $h$ on the positive side and $-h$ on the negative side. The remainder need not have the same formula on both sides.

A common pitfall is to confuse the derivative value $f'(a)$ with the function value $f(a)$. They have different roles: $f(a)$ is the value at the fixed input, while $f'(a)$ is the coefficient describing the leading change in the function for a small input increment. In the first-order approximation, these appear in separate terms, $f(a)$ and $f'(a)h$.

Another pitfall is to check only one side of the point when the definition requires a two-sided limit. At an interior point, increments can approach zero through positive and negative values. Both sides must be consistent with the same coefficient. The worked example with $u$ succeeds because both quotient expressions approach $1$; a one-sided calculation alone would not establish differentiability.

Finally, the approximation statement is local. It concerns sufficiently small increments around one fixed point, not all input values in the interval. It does not, by itself, say that the derivative function is continuous, or that the same approximation works with one fixed error bound at every point. It identifies exactly the first-order behavior at the point under consideration.

Check Your Understanding

Use the pointwise definition and the first-order approximation characterization to answer these questions.

What must be true of the difference quotient for $f$ to be differentiable at an interior point $a$?
For $g(x)=x^4$, what does the direct difference-quotient calculation give for $g'(a)$, and what is the derivative function?
For the piecewise function $u$ in the worked example, what are the positive- and negative-increment quotients at zero?
Why is the condition $r(h)/h\to0$ more informative than merely requiring $r(h)\to0$?
Why can two different coefficients not both give first-order approximations with remainders negligible compared with $h$?

What You'll Learn