The Derivative

The limit of difference quotients, read as an instantaneous rate and as the slope of the best linear approximation. The rules for powers, products, quotients and compositions, each a consequence of that limit rather than a convention, and the three ways the limit can fail to exist.

Definition

The derivative of $f$ at $a$ is

f^{'} (a) = lim_{h \to 0} \frac{f (a + h) - f (a)}{h},

when the limit exists, in which case $f$ is differentiable at $a$ . The quotient before the limit is the slope of the secant through $(a, f (a))$ and $(a + h, f (a + h))$ ; the limit is the slope of the tangent.

Two readings. As a rate, $f^{'} (a)$ is the instantaneous rate of change of $f$ per unit change in its input. As a linearisation, the tangent line $L (x) = f (a) + f^{'} (a) (x - a)$ is the unique line agreeing with $f$ to first order at $a$ : the error $f (x) - L (x)$ vanishes faster than $x - a$ .

The rules. For differentiable $f$ and $g$ :

	Rule
Power	$\frac{d}{d x} x^{n} = n x^{n - 1}$ , for every real $n$
Constant multiple	$(c f)^{'} = c f^{'}$
Sum	$(f + g)^{'} = f^{'} + g^{'}$
Product	$(f g)^{'} = f^{'} g + f g^{'}$
Quotient	${(\frac{f}{g})}^{'} = \frac{f^{'} g - f g^{'}}{g^{2}}$ , where $g \neq 0$
Chain	$(f \circ g)^{'} (x) = f^{'} (g (x)) g^{'} (x)$

The sum rule is the only one that distributes as naively expected. The product rule is not $f^{'} g^{'}$ , and the quotient rule is not $f^{'} / g^{'}$ . Both follow from the limit by adding and subtracting a cross term, not from any algebraic analogy.

Differentiability implies continuity. If $f^{'} (a)$ exists then $f$ is continuous at $a$ , because $f (a + h) - f (a) = h \cdot \frac{f (a + h) - f (a)}{h} \to 0 \cdot f^{'} (a) = 0$ . The converse fails, which is what the non-differentiable cases record.

Three ways the limit fails. A corner, where the one-sided limits exist and differ, as for $| x |$ at 0. A vertical tangent, where the quotient diverges, as for $x^{1 / 3}$ at 0. A discontinuity, where $f$ is not continuous and so cannot be differentiable.

What a zero derivative establishes. $f^{'} (c) = 0$ makes $c$ a critical point, which is necessary for an interior extremum but not sufficient: $x^{3}$ has $f^{'} (0) = 0$ and no extremum there. Deciding which case holds requires the sign of $f^{'}$ around $c$ , or the sign of $f^{″} (c)$ when it is nonzero.

Assumptions and scope

Differentiability is a property at a point, not of a function globally. $| x |$ is differentiable everywhere except 0, and saying it is 'not differentiable' without naming the point is imprecise.
Differentiable implies continuous; continuous does not imply differentiable. A corner is the standard counterexample, and functions exist that are continuous everywhere and differentiable nowhere.
The product rule is $f^{'} g + f g^{'}$ , not $f^{'} g^{'}$ . The two agree almost nowhere: for $x^{2}$ and $x^{3}$ at $x = 2$ they give 80 and 48.
$f^{'} (c) = 0$ is necessary but not sufficient for an interior extremum. $x^{3}$ at 0 is the standard failure, and the second derivative test is silent whenever $f^{″} (c) = 0$ .
The power rule holds for every real exponent, but the derivation from the binomial theorem covers only positive integers. The general case needs the chain rule with logarithms, or a limit argument.
A one-sided derivative may exist where the two-sided one does not. Endpoints of a closed interval admit only the one-sided notion.

Worked material

Example

Derivatives of the functions that keep appearing

Each of these is worth knowing on sight, and each says something the formula alone does not.

A constant, $f (x) = 7$ . The difference quotient is $\frac{7 - 7}{h} = 0$ for every $h$ , so $f^{'} (x) = 0$ everywhere. A constant function has no rate of change, which is the base case the sum rule leans on whenever a constant term is dropped.

A line, $f (x) = 3 x + 2$ . The quotient is $\frac{3 (x + h) + 2 - 3 x - 2}{h} = \frac{3 h}{h} = 3$ , with no limit needed. It is already free of $h$ . The derivative is the slope, constant everywhere, and the tangent to a line is the line itself. This is the case where the linear approximation is exact rather than merely good nearby.

A square, $f (x) = x^{2}$ . $f^{'} (x) = 2 x$ . The derivative is negative for $x < 0$ , zero at the vertex, positive for $x > 0$ . The parabola falling, levelling, then rising. Reading the sign of $f^{'}$ off the graph's shape, and the shape off the sign, is the habit this unit is building.

A reciprocal, $f (x) = 1 / x = x^{- 1}$ . The power rule gives $f^{'} (x) = - x^{- 2} = - 1 / x^{2}$ , negative for every $x \neq 0$ : the function decreases on both branches. At $x = 4$ , $f^{'} (4) = - 1 / 16$ , a gentle slope; at $x = 0.1$ it is $- 100$ , extremely steep. The derivative grows without bound near the origin, which is the analytic form of the vertical asymptote.

A root, $f (x) = \sqrt{x} = x^{1 / 2}$ . The power rule applies with a fractional exponent: $f^{'} (x) = \frac{1}{2} x^{- 1 / 2} = \frac{1}{2 \sqrt{x}}$ . At $x = 4$ that is $\frac{1}{4}$ ; at $x = 100$ it is $\frac{1}{20}$ . The slope decreases as $x$ grows, the curve flattens, and as $x \to 0^{+}$ it diverges, giving the vertical tangent at the origin. Note that $f$ is defined at 0 but not differentiable there.

A cubic with a flat spot, $f (x) = x^{3}$ . $f^{'} (x) = 3 x^{2} \geq 0$ everywhere, and zero only at $x = 0$ . The function is increasing throughout, yet its tangent at the origin is horizontal. This is the standard counterexample to the belief that a vanishing derivative signals a turning point.

Four of the six are instances of the single power rule, with exponents $0$ , $1$ , $2$ , $- 1$ , $\frac{1}{2}$ and $3$ . One formula covering constants, lines, curves, reciprocals and roots. The two that behave unusually, $1 / x$ near 0 and $\sqrt{x}$ at 0, are unusual for the same reason: the derivative diverges where the graph turns vertical.

Non-example

Where the derivative fails to exist

Three failures of the limit.

A corner: $f (x) = | x |$ at $x = 0$ . The difference quotient is $\frac{| h |}{h}$ , which equals $+ 1$ for every $h > 0$ and $- 1$ for every $h < 0$ . Both one-sided limits exist and they disagree, so the two-sided limit does not. The function is continuous at 0. This is the standard demonstration that continuity does not imply differentiability.

A vertical tangent: $f (x) = x^{1 / 3}$ at $x = 0$ . The quotient is $\frac{h^{1 / 3}}{h} = h^{- 2 / 3}$ , which grows without bound: $100$ at $h = 10^{- 3}$ , $10,000$ at $h = 10^{- 6}$ . Here the one-sided limits agree, both are $+ \infty$ , but agreeing on $\infty$ is not having a limit, since no real number is approached. The tangent line exists geometrically and is vertical, which has no finite slope.

A discontinuity. Any $f$ discontinuous at $a$ fails there, by the contrapositive of "differentiable implies continuous". A jump is the clearest case: the numerator $f (a + h) - f (a)$ does not tend to 0, so the quotient diverges.

Rules that do not hold.

$(f g)^{'} \neq f^{'} g^{'}$ . With $f = x^{2}$ and $g = x^{3}$ at $x = 2$ : the product rule gives $4 \cdot 8 + 4 \cdot 12 = 80$ , and the simplification $f g = x^{5}$ confirms $5 \cdot 16 = 80$ . The false rule gives $4 \cdot 12 = 48$ . The error is not small and does not vanish for large $x$ .

${(\frac{f}{g})}^{'} \neq \frac{f^{'}}{g^{'}}$ . For $\frac{x^{2} + 1}{x - 1}$ at $x = 3$ the quotient rule gives $\frac{1}{2}$ , while $\frac{f^{'}}{g^{'}} = \frac{6}{1} = 6$ , wrong by a factor of twelve.

The chain rule's inner factor is not optional. For $(3 x^{2} + 1)^{4}$ at $x = 1$ , the correct value is $4 \cdot 64 \cdot 6 = 1536$ ; omitting $u^{'} = 6 x$ gives 256.

Inferences the derivative does not license.

$f^{'} (c) = 0$ does not make $c$ an extremum. $f (x) = x^{3}$ has $f^{'} (0) = 0$ , yet $x^{3}$ is strictly increasing everywhere and 0 is neither a maximum nor a minimum. It is an inflection point with a horizontal tangent.

$f^{″} (c) = 0$ decides nothing. At $x = 0$ the functions $x^{3}$ , $x^{4}$ and $- x^{4}$ all have vanishing first and second derivatives, and have respectively no extremum, a minimum and a maximum. The second-derivative test is silent here, and the sign of $f^{'}$ on either side must be used instead.

A local extremum is not a global one. $x^{3} - 3 x$ has a local maximum at $x = - 1$ with value 2, while the function exceeds 2 for all $x > 2$ , indeed $f (3) = 18$ . Nothing about a derivative at a point constrains behaviour far from it.

Common errors

Common misconception

The derivative of a product is the product of the derivatives, so $(f g)^{'} = f^{'} g^{'}$ .

Related units

Connected

Linear Transformations (related)
Quadratic Forms and Definiteness (related)

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The Derivative

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Calculus

Sources

Calculus: Early Transcendentals (2020)

The Derivative

Assumptions and scope

Worked material

Derivatives of the functions that keep appearing

Where the derivative fails to exist

Common errors

Related units

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