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Topos Uranos
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Higher-order derivatives and the differential

Summary

This lesson differentiates the derivative. The derivative of a function is another function, which can in turn be differentiated: in this way one obtains the second derivative, the third and, in general, the derivative of order nn, which for powers, the exponential, sine and cosine has a closed formula, proved by induction. The derivative of order nn of a product follows the same law as the power of a binomial: it is the Leibniz rule. The second half of the lesson gives meaning, separately, to the signs dy\mathrm{d}y and dx\mathrm{d}x of Leibniz notation: the differential dy=f′(x) dx\mathrm{d}y = f'(x) \, \mathrm{d}x is the linear part of the increment of the function, the only one that differs from it by a quantity negligible compared with the increment of the variable. With it one approximates values and estimates how the error of a measurement propagates, without yet attempting to bound the error of the approximation.

Learning objectives

  1. Define the higher-order derivatives, with their domain, and obtain by induction the derivative of order nn of powers, of the exponential, of sine, of cosine and of polynomials.
  2. Prove the Leibniz rule for the derivative of order nn of a product and apply it when one of the factors is a polynomial.
  3. Define the differential and prove that it is the principal part of the increment, distinguishing Δy\Delta y from dy\mathrm{d}y.
  4. Approximate values and estimate relative errors with the differential, knowing what the approximation asserts and what it does not.

Higher-order derivatives

The derivative of a function is another function, and nothing prevents differentiating it again. The derivative of x3x^3 is 3x23x^2, and that of 3x23x^2 is 6x6x; that of 6x6x is 66, and that of 66 is 00. If s(t)s(t) is the position of a moving point at the instant tt, its derivative s′(t)s'(t) is the velocity, and the derivative of the velocity, which measures how fast the velocity itself changes, is the acceleration; the lesson on rates of change will study that reading. As in the lesson on the derivative as a limit, it is worth fixing precisely where each derivative is defined.

DefinitionHigher-order derivatives

Let ff be a function defined on AA. The higher-order derivatives of ff are defined by recursion: the derivative of order 00 is the function itself, and the derivative of order n+1n + 1 is the derivative function of the one of order nn:

f(0)=f,f(n+1)=(f(n))′(n≥0)f^{(0)} = f, \qquad f^{(n + 1)} = \left(f^{(n)}\right)' \quad (n \geq 0)

By the definition of the derivative function, the domain of the derivative of order n+1n + 1 is the set of points of the domain of f(n)f^{(n)} at which f(n)f^{(n)} is differentiable. One says that ff is nn times differentiable at a point aa if aa belongs to the domain of f(n)f^{(n)}; on a set SS, if it is so at each point of SS; and infinitely differentiable on SS, if it is so for every nn. The first ones are written with primes, f′′=f(2)f'' = f^{(2)} (the second derivative) and f′′′=f(3)f''' = f^{(3)}; from order four on, with the order in parentheses, as in f(4)f^{(4)}, so as not to confuse it with a power. If y=f(x)y = f(x), in Leibniz notation one writes d2ydx2\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} and dnydxn\frac{\mathrm{d}^n y}{\mathrm{d}x^n}, and the operation of differentiating nn times, dndxn\frac{\mathrm{d}^n}{\mathrm{d}x^n}.

In Leibniz notation, the exponent in the numerator indicates how many times the operation ddx\frac{\mathrm{d}}{\mathrm{d}x} is applied, and the one in the denominator affects the whole symbol dx\mathrm{d}x, not the variable xx; the complete sign d2ydx2\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} is read as a whole. If ff is nn times differentiable at a point, each derivative of order less than nn is differentiable there and, by the theorem from the lesson on the derivative as a limit stating that differentiable implies continuous, it is continuous there. The second derivative may be missing even though the first exists: the lesson on differentiation rules showed that x∣x∣x \abs{x} is differentiable on all of R\mathbb{R}, with derivative 2∣x∣2\abs{x}, and the absolute value is not differentiable at 00, by the proposition on continuity without a derivative. Hence x∣x∣x \abs{x} is differentiable once on R\mathbb{R} and twice only on R∖{0}\mathbb{R} \setminus \{0\}, where its second derivative equals 22 to the right of 00 and −2-2 to the left.

To avoid repeating hypotheses about the domain, the functions in the general statements are taken to be defined on an open interval II, all of whose points are accumulation points of II. Nothing is lost: by the second part of the lemma on the local character of the derivative, from the lesson on differentiation rules, at a point of an open interval contained in the domain the derivative of a function is that of its restriction to the interval, and, by induction, the same holds for the higher-order derivatives. Thus 1x\frac{1}{x} is studied on (−∞,0)\ioo{-\infty}{0} and on (0,∞)\ioo{0}{\infty}.

LemmaLinearity of higher-order derivatives

Let ff and gg be functions defined on an open interval II and nn times differentiable on II, and let α\alpha be a real number. Then h=αf+gh = \alpha f + g is nn times differentiable on II, and h(n)=αf(n)+g(n)h^{(n)} = \alpha f^{(n)} + g^{(n)} on II. The theorem on the derivative of a linear combination, from the lesson on differentiation rules, is applied nn times.

Proof

  1. h′(x)=αf′(x)+g′(x)h'(x) = \alpha f'(x) + g'(x)

    For n=1n = 1 it is the theorem on the derivative of a linear combination, at each point of II.

  2. h(n+1)(x)=αf(n+1)(x)+g(n+1)(x)h^{(n + 1)}(x) = \alpha f^{(n + 1)}(x) + \resaltar{g^{(n + 1)}(x)}

    If the equality holds for nn, and ff and gg are n+1n + 1 times differentiable on II, the function h(n)=αf(n)+g(n)h^{(n)} = \alpha f^{(n)} + g^{(n)} is a linear combination of two functions differentiable on II; by the same theorem, it is differentiable on II. By the principle of induction, the equality holds for every nn.