Generalization of the product rule in calculus
In calculus, the general Leibniz rule,[1] named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if
and
are
-times differentiable functions, then the product
is also
-times differentiable and its
th derivative is given by
![{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(n-k)}g^{(k)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b5fbf529aa458b37f32e4cf1839132d83af06e8)
where
is the binomial coefficient and
denotes the jth derivative of f (and in particular
).
The rule can be proved by using the product rule and mathematical induction.
Second derivative
If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions:
![{\displaystyle (fg)''(x)=\sum \limits _{k=0}^{2}{{\binom {2}{k}}f^{(2-k)}(x)g^{(k)}(x)}=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b891fa0132b569683c94a9fbd3a32c025996911)
More than two factors
The formula can be generalized to the product of m differentiable functions f1,...,fm.
![{\displaystyle \left(f_{1}f_{2}\cdots f_{m}\right)^{(n)}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{n \choose k_{1},k_{2},\ldots ,k_{m}}\prod _{1\leq t\leq m}f_{t}^{(k_{t})}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ddac51dd078e1e5f095a0e67c853c6bb278bf0)
where the sum extends over all m-tuples (k1,...,km) of non-negative integers with
and
![{\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7165fdb93f8d28ab738a85570ce10529dcdad8)
are the multinomial coefficients. This is akin to the multinomial formula from algebra.
Proof
The proof of the general Leibniz rule proceeds by induction. Let
and
be
-times differentiable functions. The base case when
claims that:
![{\displaystyle (fg)'=f'g+fg',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c58d30b80dd780ca4bd593f0d1960ae928113562)
which is the usual product rule and is known to be true. Next, assume that the statement holds for a fixed
that is, that
![{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}g^{(k)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac28152884f452b42b5bcf412edb7f0ad8635d39)
Then,
![{\displaystyle {\begin{aligned}(fg)^{(n+1)}&=\left[\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}g^{(k)}\right]'\\&=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n+1-k)}g^{(k)}+\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}g^{(k+1)}\\&=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n+1-k)}g^{(k)}+\sum _{k=1}^{n+1}{\binom {n}{k-1}}f^{(n+1-k)}g^{(k)}\\&={\binom {n}{0}}f^{(n+1)}g+\sum _{k=1}^{n}{\binom {n}{k}}f^{(n+1-k)}g^{(k)}+\sum _{k=1}^{n}{\binom {n}{k-1}}f^{(n+1-k)}g^{(k)}+{\binom {n}{n}}fg^{(n+1)}\\&={\binom {n+1}{0}}f^{(n+1)}g+\left(\sum _{k=1}^{n}\left[{\binom {n}{k-1}}+{\binom {n}{k}}\right]f^{(n+1-k)}g^{(k)}\right)+{\binom {n+1}{n+1}}fg^{(n+1)}\\&={\binom {n+1}{0}}f^{(n+1)}g+\sum _{k=1}^{n}{\binom {n+1}{k}}f^{(n+1-k)}g^{(k)}+{\binom {n+1}{n+1}}fg^{(n+1)}\\&=\sum _{k=0}^{n+1}{\binom {n+1}{k}}f^{(n+1-k)}g^{(k)}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6541f33651e7db5b47a206be93b1cdb3930c6fa5)
And so the statement holds for
and the proof is complete.
Multivariable calculus
With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally:
![{\displaystyle \partial ^{\alpha }(fg)=\sum _{\beta \,:\,\beta \leq \alpha }{\alpha \choose \beta }(\partial ^{\beta }f)(\partial ^{\alpha -\beta }g).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5c0da3e788d6e7e6d23af152f454a485c77a47)
This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let P and Q be differential operators (with coefficients that are differentiable sufficiently many times) and
Since R is also a differential operator, the symbol of R is given by:
![{\displaystyle R(x,\xi )=e^{-{\langle x,\xi \rangle }}R(e^{\langle x,\xi \rangle }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be6a3478fb3d358beff45a19c0eff689bbca23f8)
A direct computation now gives:
![{\displaystyle R(x,\xi )=\sum _{\alpha }{1 \over \alpha !}\left({\partial \over \partial \xi }\right)^{\alpha }P(x,\xi )\left({\partial \over \partial x}\right)^{\alpha }Q(x,\xi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fb38e11e39df02c5d28b4c344f450a0609d504)
This formula is usually known as the Leibniz formula. It is used to define the composition in the space of symbols, thereby inducing the ring structure.
See also
References