Leibniz rule

2020 Mathematics Subject Classification: Primary: 26A06 Secondary: 26B05 [MSN][ZBL]

2020 Mathematics Subject Classification: Primary: 58A05 [MSN][ZBL]

2020 Mathematics Subject Classification: Primary: 12H05 [MSN][ZBL]

product rule

In calculus, the term refers to the elementary rule for the derivative of the product of two functions. In its simplest form it states the following

Theorem 1 Let $I$ be an open interval and $x_0\in I$. Let $f,g : I \to \mathbb R$ be two functions which are differentiable at $x_0$. Then $fg$ is also differentiable at $x_0$ and \begin{equation}\label{e:rule} (fg)' (x_0) = f(x_0) g'(x_0) + f' (x_0) g (x_0)\, . \end{equation}

The rule can be easily extended with an inductive procedure to higher derivatives. Under the assumptions that both functions are $n$ times differentiable at $x_0$, we have \[ (fg)^{(n)} (x_0) = \sum_{k=0}^n {n \choose k}\, f^{(n-k)} (x_0)\, g^{(k)} (x_0)\, . \]

Variants

The same rule applies also to several other situations. The following is a list of rather important cases.

• $x_0\in U$ open subset of $\mathbb R^n$ and the maps $f,g: U \to \mathbb R$ have both the partial derivative $\frac{\partial}{\partial x_i}$ at $x_0$. The corresponding formula is then

\[ \frac{\partial}{\partial x_i} (fg) (x_0) = g(x_0) \frac{\partial f}{\partial x_i} (x_0) + f(x_0) \frac{\partial g}{\partial x_i} (x_0)\, . \]

• $x_0\in U$ open subset of $\mathbb R^n$, or more generally $x_0\in M$ for some differentiable manifold, and the maps $f,g U \to \mathbb R$ are differentiable along a vector field. The rule becomes then

\[ [X (fg)]\, (x_0) = g (x_0)\, [X (f)]\, (x_0) + f(x_0)\, [X (g)]\, (x_0)\, . \]

• $x_0\in U$ open subset of $\mathbb R^n$, or more generally $x_0\in M$ for some differentiable manifold, and the maps $f,g: U \to \mathbb R$ are differentiable. The formula is then

\[ \left. d (fg) \right|_{x_0} = f(x_0)\, \left. dg\right|_{x_0} + g(x_0)\, \left. df\right|_{x_0}\, . \]

• $z_0\in U$ open subset of $\mathbb C$ and $f,g: U \to \mathbb C$ are differentiable in the sense of complex analysis (cf. Analytic function). Then the formula reads as \eqref{e:rule}.

Algebraic generalizations

The Leibniz rule is, together with the linearity, the key algebraic identity which unravels most of the structural properties of the differentiation. For this reason, in several situations people call derivations those operations over an appropriate set of functions which are linear and satisfy the Leibniz rule. Two important instances are:

• Derivation in a ring. The primary example is the following: given a differentiable manifold $M$, consider the ring $C^1 (M)$ of $C^1$ real-valued functions over $M$. A derivation at $x_0$ is a map $D: C^1 (M) \to \mathbb R$ which is linear, i.e.

\[ D (\lambda f + \mu g) = \lambda Df + \mu Dg \qquad \forall \lambda, \mu \in \mathbb R, \forall f,g\in C^1 (M)\, , \] and satisfies the Leibniz rule, i.e. \[ D (fg) = f(x_0) Dg + g (x_0) Df\, . \] Global derivatives are maps from $C^1 (M)$ to $C^0 (M)$ satisfying the (analog of) the same rules. Derivations give one way (among many equivalent others) to define the tangent space to differentiable manifolds and vector fields on them (together with the Lie bracket).

• Differential field. A field $\mathbb F$ with a map $':\mathbb F \to \mathbb F$ (called derivation) satisfying the rules

\[ (f+g)' = f' + g' \qquad \mbox{and} \qquad (fg)' = fg' + gf'\, . \] Differential fields are the object of study of Differential algebra. Historically the birth of differential algebra can be dated back to the following famous theorem of Liouville ([Li]): the primitive of $e^{x^2}$ cannot be expressed in terms of elementary functions (see [Ro] for a modern proof).

References

[Ca]	H. Cartan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995).
[Li]	J. Liouville, "Mémoire sur les Trascendantes Elliptiques et sur l’impossibilité d’exprimer les racines de certaines équations en fonction finie explicite des coefficients", J. Math. Pures Appl. 2 (1837) pp. 124–193.
[Ma]	A. R. Magid, "Lectures on differential Galois theory", University Lecture Series, 7. American Mathematical Society, Providence, RI, (1994)
[Ro]	M. Rosenlicht, "Integration in finite terms", Amer. Math. Monthly 79 (1972) pp. 963–972.
[Ru]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) MR0385023 Zbl 0346.26002
[Sp]	M. Spivak, "Calculus on manifolds" , Benjamin (1965)