Differentiation and Derivatives

In §Continuous Functions, we reformulated what it means for a function to be continuous in the previously introduced language of \(\epsilon-\delta\). Now the natural next step is to define the derivative of a function.

Definition of the Derivative

The slope of the line connecting two points \((a, f(a))\) and \((x, f(x))\) is the average rate of change \(\frac{f(x) - f(a)}{x - a}\), and taking the limit as \(x\) approaches \(a\) yields the derivative.

Definition 1 A function \(f\) is said to be differentiable at a point \(a\) if the limit

\[f'(a) := \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a} = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h}\]

exists. This value \(f'(a)\) is called the derivative of \(f\) at \(a\). If \(f\) is differentiable at every point of its domain, the function defined by \(a \mapsto f'(a)\), denoted \(f'\), is called the derivative function of \(f\).

The two expressions are, of course, trivially the same upon setting \(x=a+h\), and by the discussion of the average rate of change above, the derivative \(f'(a)\) is the slope of the tangent line to the graph at the point \((a, f(a))\). On the other hand, any line is completely determined by its slope and one point on it; in particular, since it is already given that this tangent line passes through \((a,f(a))\), the tangent line to a function differentiable at \(a\) is given by

\[y = f(a) + f'(a)(x - a)\]

Near \(a\), this linear function best approximates \(f\), so when \(h\) is small, we write \(f(a+h) \approx f(a) + f'(a)h\). The derivative is also written following Leibniz as

\[\frac{\dd{f}}{\dd{x}},\qquad \frac{d}{\dd{x}}f\]

and so on, but in calculus the \(f'\) notation is often sufficient.

Applying the definition directly, for instance, the average rate of change of \(f(x) = x^2\) is \(\frac{(a+h)^2 - a^2}{h} = 2a + h\), so \(f'(a) = 2a\), and the derivative of a constant function is always \(0\). In the same way, for \(f(x) = 1/x\) (\(x \neq 0\)), finding a common denominator for the average rate of change yields \(f'(a) = -1/a^2\), and for \(f(x) = \sqrt{x}\) (\(x > 0\)), rationalizing the numerator yields \(f'(a) = 1/(2\sqrt{a})\).

Differentiability and Continuity

Differentiability is a stronger condition than continuity.

Proposition 2 If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

Proof

For \(x \neq a\),

\[f(x) - f(a) = \frac{f(x)-f(a)}{x-a}\cdot(x-a)\]

As \(x \rightarrow a\), the first factor on the right converges to \(f'(a)\) and the second factor converges to \(0\), so by §Limits of Functions, ⁋Proposition 5 (Limit Laws),

\[\lim_{x\rightarrow a}\bigl(f(x)-f(a)\bigr) = f'(a)\cdot 0 = 0\]

Therefore \(f\) is continuous at \(a\).

However, the converse does not hold. A representative example is \(f(x) = \lvert x\rvert\), which is continuous at \(0\) but not differentiable there.

Indeed,

\[\frac{f(h)-f(0)}{h} = \frac{\lvert h\rvert}{h}=\begin{cases}1&\text{if $h>0$}\\-1&\text{if $h<0$}\end{cases}\]

so the limit of the average rate of change of this function is \(1\) as \(h \rightarrow 0^+\) and \(-1\) as \(h \rightarrow 0^-\); since these differ, the limit does not exist.

As a similar example, \(f(x) = \sqrt[3]{x}\) has a vertical tangent at \(0\), where the average rate of change diverges as \(h^{-2/3} \rightarrow \infty\).

On the other hand, \(f(x) = x^{2/3}\) forms a cusp at \(0\), where the left and right average rates of change diverge to \(\mp\infty\).

These are examples that fail to be differentiable only at one or two points, but nondifferentiability can be far worse: there even exist functions like the Weierstrass function that are continuous on all of \(\mathbb{R}\) yet have no tangent line at any point.

Conversely, differentiability is a local property that may hold at only a single point, so an extreme example of the following form is also possible.

Example 3 (A function differentiable at only one point) Consider the function

\[f(x) = \begin{cases} x^2 & (x \in \mathbb{Q}) \\ 0 & (x \notin \mathbb{Q}) \end{cases}\]

For \(a \neq 0\), along a sequence of rationals and a sequence of irrationals converging to \(a\), the function values split into \(a^2\) and \(0\) respectively, so \(f\) is discontinuous, and by the contrapositive of Proposition 2 it is not differentiable. On the other hand, at \(0\) we have \(\lvert f(x)\rvert \leq x^2\), so \(f\) is continuous, and since the average rate of change satisfies

\[\left\lvert \frac{f(x) - f(0)}{x - 0} \right\rvert = \frac{\lvert f(x)\rvert}{\lvert x\rvert} \leq \lvert x\rvert \rightarrow 0\]

\(f'(0) = 0\) exists. That is, \(f\) is differentiable only at \(0\).

Properties of Differentiation

Meanwhile, since the derivative is ultimately the limit of a function, namely the average rate of change, we can prove from §Limits of Functions, ⁋Proposition 5 (Limit Laws) that differentiation also possesses linearity.

Proposition 4 If \(f, g\) are differentiable at \(a\) and \(c\) is a constant, then \(f + g\) and \(cf\) are also differentiable at \(a\), and

\[(f+g)'(a) = f'(a) + g'(a), \qquad (cf)'(a) = c f'(a)\]

Proof

The average rate of change splits as

\[\frac{(f+g)(a+h)-(f+g)(a)}{h} = \frac{f(a+h)-f(a)}{h} + \frac{g(a+h)-g(a)}{h}\]

and since each term converges to \(f'(a), g'(a)\), the sum also converges by §Limits of Functions, ⁋Proposition 5 (Limit Laws). Examining the average rate of change expression for \(cf\) yields the same result.

On the other hand, since the derivative \(f'\) is itself a function, if it is differentiable we can differentiate it again.

Definition 5 If the derivative of \(f\), \(f'\), is differentiable, its derivative is called the second derivative \(f'' = (f')'\). Repeating this, the result of differentiating \(n\) times is called the \(n\)-th derivative \(f^{(n)}\), and we set \(f^{(0)} = f\). If \(f^{(n)}\) exists and is continuous on an interval, we say \(f\) is of class \(C^n\) on that interval, and if derivatives of all orders exist, we say \(f\) is of class \(C^\infty\) or smooth.

The second derivative \(f''\) is the rate of change of the rate of change; if \(f\) is position, then \(f''\) is acceleration. Also, the sign of \(f''\) indicates the direction in which the graph bends (convexity/concavity). In this post we do not aim to differentiate specific functions, but in the next post we will see that polynomial functions and functions such as \(\sin, \cos, \exp\) are smooth.

Finally, just as a limit exists only when the left and right limits agree, the existence of the derivative also depends on whether the limits of the left and right average rates of change agree. Defining these separately allows us to handle differentiability at corners and endpoints precisely.

Definition 6 The right-hand derivative and left-hand derivative of a function \(f\) at a point \(a\) are defined respectively by

\[f'_+(a) := \lim_{h \rightarrow 0^+} \frac{f(a+h) - f(a)}{h}, \qquad f'_-(a) := \lim_{h \rightarrow 0^-} \frac{f(a+h) - f(a)}{h}\]

Both one-sided derivatives exist and are equal if and only if \(f\) is differentiable at \(a\), in which case their common value is \(f'(a)\). This is nothing more than applying to the average rate of change the fact that a limit exists only when both one-sided limits agree. (§Limits of Functions) Revisiting the absolute value function \(f(x) = \lvert x\rvert\), we have \(f'_+(0) = 1\) and \(f'_-(0) = -1\), which differ, so nondifferentiability at \(0\) is immediately confirmed. One-sided derivatives are also used naturally when discussing differentiability at an endpoint of the domain; for instance, for the function defined on \([0, \infty)\) by \(f(x) = \sqrt{x}\), differentiability at \(0\) is meaningful only via the right-hand derivative.

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References

[Ste] J. Stewart, Calculus, 8th ed., Cengage Learning, 2016.
[Kim] 김홍종, 미적분학 1·2, 제3개정판, 서울대학교출판문화원, 2020.