Product Rule
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Product Rule

Geometric illustration of a proof of the product rule

In calculus, the product rule (or Leibniz rule[1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as

${\displaystyle (u\cdot v)'=u'\cdot v+u\cdot v'}$
or in Leibniz's notation as
${\displaystyle {\frac {d}{dx}}(u\cdot v)={\frac {du}{dx}}\cdot v+u\cdot {\frac {dv}{dx}}.}$

The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts.

## Discovery

Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials.[2] (However, J. M. Child, a translator of Leibniz's papers,[3] argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is

{\displaystyle {\begin{aligned}d(u\cdot v)&{}=(u+du)\cdot (v+dv)-u\cdot v\\&{}=u\cdot dv+v\cdot du+du\cdot dv.\end{aligned}}}

Since the term du·dv is "negligible" (compared to du and dv), Leibniz concluded that

${\displaystyle d(u\cdot v)=v\cdot du+u\cdot dv}$

and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain

${\displaystyle {\frac {d}{dx}}(u\cdot v)=v\cdot {\frac {du}{dx}}+u\cdot {\frac {dv}{dx}}}$

which can also be written in Lagrange's notation as

${\displaystyle (u\cdot v)'=v\cdot u'+u\cdot v'.}$

## Examples

• Suppose we want to differentiate f(x) = x2 sin(x). By using the product rule, one gets the derivative f(x) = 2x sin(x) + x2 cos(x) (since the derivative of x2 is 2x and the derivative of the sine function is the cosine function).
• One special case of the product rule is the constant multiple rule, which states: if c is a number and f(x) is a differentiable function, then cf(x) is also differentiable, and its derivative is (cf)(x) = cf(x). This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear.
• The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable, but only says what its derivative is if it is differentiable.)

## Proofs

### Proof by factoring (from first principles)

Let h(x) = f(x)g(x) and suppose that f and g are each differentiable at x. We want to prove that h is differentiable at x and that its derivative, h(x), is given by f(x)g(x) + f(x)g(x). To do this, ${\displaystyle f(x)g(x+\Delta x)-f(x)g(x+\Delta x)}$ (which is zero, and thus does not change the value) is added to the numerator to permit its factoring, and then properties of limits are used.

{\displaystyle {\begin{aligned}h'(x)&=\lim _{\Delta x\to 0}{\frac {h(x+\Delta x)-h(x)}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)g(x+\Delta x)-f(x)g(x)}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)g(x+\Delta x)-f(x)g(x+\Delta x)+f(x)g(x+\Delta x)-f(x)g(x)}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {{\big [}f(x+\Delta x)-f(x){\big ]}\cdot g(x+\Delta x)+f(x)\cdot {\big [}g(x+\Delta x)-g(x){\big ]}}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}\cdot \underbrace {\lim _{\Delta x\to 0}g(x+\Delta x)} _{\text{See the note below.}}+\lim _{\Delta x\to 0}f(x)\cdot \lim _{\Delta x\to 0}{\frac {g(x+\Delta x)-g(x)}{\Delta x}}\\[5pt]&=f'(x)g(x)+f(x)g'(x).\end{aligned}}}

The fact that

${\displaystyle \lim _{\Delta x\to 0}g(x+\Delta x)=g(x)}$

is deduced from a theorem that states that differentiable functions are continuous.

### Brief proof

By definition, if ${\displaystyle f,g:\mathbb {R} \to \mathbb {R} }$ are differentiable at ${\displaystyle x}$ then we can write

${\displaystyle f(x+h)=f(x)+f'(x)h+\psi _{1}(h)}$
and
${\displaystyle g(x+h)=g(x)+g'(x)h+\psi _{2}(h)}$
such that ${\textstyle \lim _{h\to 0}{\frac {\psi _{1}(h)}{h}}=\lim _{h\to 0}{\frac {\psi _{2}(h)}{h}}=0,}$ also written ${\displaystyle \psi _{1},\psi _{2}\sim o(h^{3})}$. Then:
{\displaystyle {\begin{aligned}f(x+h)g(x+h)-f(x)g(x)&=(f(x)+f'(x)h+\psi _{1}(h))(g(x)+g'(x)h+\psi _{2}(h))-f(x)g(x)\\&=f(x)g(x)+f'(x)g(x)h+f(x)g'(x)h-f(x)g(x)+{\text{other terms}}\\&=f'(x)g(x)h+f(x)g'(x)h+o(h^{3})\\[12pt]\end{aligned}}}
The "other terms" consist of items such as ${\displaystyle f(x)\psi _{2}(h),f'(x)g'(x)h^{2}}$ and ${\displaystyle hf'(x)\psi _{1}(h).}$ It is not difficult to show that they are all ${\displaystyle o(h^{3}).}$ Dividing by ${\displaystyle h}$ and taking the limit for small ${\displaystyle h}$ gives the result.

### Quarter squares

There is a proof using quarter square multiplication which relies on the chain rule and on the properties of the quarter square function (shown here as q, i.e., with ${\displaystyle q(x)={\tfrac {x^{2}}{4}}}$).

Let

${\displaystyle f=q(u+v)-q(u-v),}$

Differentiating both sides:

{\displaystyle {\begin{aligned}f'&=q'(u+v)(u'+v')-q'(u-v)(u'-v')\\[4pt]&=\left({1 \over 2}(u+v)(u'+v')\right)-\left({1 \over 2}(u-v)(u'-v')\right)\\[4pt]&={1 \over 2}(uu'+vu'+uv'+vv')-{1 \over 2}(uu'-vu'-uv'+vv')\\[4pt]&=vu'+uv'\\[4pt]&=uv'+u'v\end{aligned}}}

### Chain rule

The product rule can be considered a special case of the chain rule for several variables.

${\displaystyle {d(ab) \over dx}={\frac {\partial (ab)}{\partial a}}{\frac {da}{dx}}+{\frac {\partial (ab)}{\partial b}}{\frac {db}{dx}}=b{\frac {da}{dx}}+a{\frac {db}{dx}}.}$

### Non-standard analysis

Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a finite hyperreal number the real infinitely close to it, this gives

{\displaystyle {\begin{aligned}{\frac {d(uv)}{dx}}&=\operatorname {st} \left({\frac {(u+du)(v+dv)-uv}{dx}}\right)\\[4pt]&=\operatorname {st} \left({\frac {uv+u\cdot dv+v\cdot du+dv\cdot du-uv}{dx}}\right)\\[4pt]&=\operatorname {st} \left({\frac {u\cdot dv+(v+dv)\cdot du}{dx}}\right)\\[4pt]&=u{\frac {dv}{dx}}+v{\frac {du}{dx}}.\end{aligned}}}

This was essentially Leibniz's proof exploiting the transcendental law of homogeneity (in place of the standard part above).

### Smooth infinitesimal analysis

In the context of Lawvere's approach to infinitesimals, let dx be a nilsquare infinitesimal. Then du = u′ dx and dv = v ′ dx, so that

{\displaystyle {\begin{aligned}d(uv)&=(u+du)(v+dv)-uv\\&=uv+u\cdot dv+v\cdot du+du\cdot dv-uv\\&=u\cdot dv+v\cdot du+du\cdot dv\\&=u\cdot dv+v\cdot du\,\!\end{aligned}}}

since

${\displaystyle du\,dv=u'v'(dx)^{2}=0.}$

### Using log

Let ${\displaystyle f(x)=u(x)v(x)}$

Taking natural log on both sides,

${\displaystyle \ln f=\ln u+\ln v}$

Differentiating wrt x,

{\displaystyle {\begin{aligned}{\dfrac {f'}{f}}&={\dfrac {u'}{u}}+{\dfrac {v'}{v}}\\f'&=uv({\dfrac {u'}{u}}+{\dfrac {v'}{v}})\\f'&=u'v+uv'\end{aligned}}}

## Generalizations

### Product of more than two factors

The product rule can be generalized to products of more than two factors. For example, for three factors we have

${\displaystyle {\frac {d(uvw)}{dx}}={\frac {du}{dx}}vw+u{\frac {dv}{dx}}w+uv{\frac {dw}{dx}}.}$

For a collection of functions ${\displaystyle f_{1},\dots ,f_{k}}$, we have

${\displaystyle {\frac {d}{dx}}\left[\prod _{i=1}^{k}f_{i}(x)\right]=\sum _{i=1}^{k}\left(\left({\frac {d}{dx}}f_{i}(x)\right)\prod _{j=1,j\neq i}^{k}f_{j}(x)\right)=\left(\prod _{i=1}^{k}f_{i}(x)\right)\left(\sum _{i=1}^{k}{\frac {f'_{i}(x)}{f_{i}(x)}}\right).}$

The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative of a function f, denoted here Logder(f), is the derivative of the logarithm of the function. It follows that

${\displaystyle \operatorname {Logder} (f)={\frac {f'}{f}}.}$

Using that the logarithm of a product is the sum of the logarithms of the factors, the sum rule for derivatives gives immediately

${\displaystyle \operatorname {Logder} (f_{1}\cdots f_{k})=\sum _{i=1}^{k}\operatorname {Logder} (f_{i}).}$

The last above expression of the derivative of a product is obtained by multiplying both members of this equation by the product of the ${\displaystyle f_{i}.}$

### Higher derivatives

It can also be generalized to the general Leibniz rule for the nth derivative of a product of two factors, by symbolically expanding according to the binomial theorem:

${\displaystyle d^{n}(uv)=\sum _{k=0}^{n}{n \choose k}\cdot d^{(n-k)}(u)\cdot d^{(k)}(v).}$

Applied at a specific point x, the above formula gives:

${\displaystyle (uv)^{(n)}(x)=\sum _{k=0}^{n}{n \choose k}\cdot u^{(n-k)}(x)\cdot v^{(k)}(x).}$

Furthermore, for the nth derivative of an arbitrary number of factors:

${\displaystyle \left(\prod _{i=1}^{k}f_{i}\right)^{(n)}=\sum _{j_{1}+j_{2}+\cdots +j_{k}=n}{n \choose j_{1},j_{2},\ldots ,j_{k}}\prod _{i=1}^{k}f_{i}^{(j_{i})}.}$

### Higher partial derivatives

For partial derivatives, we have[4]

${\displaystyle {\partial ^{n} \over \partial x_{1}\,\cdots \,\partial x_{n}}(uv)=\sum _{S}{\partial ^{|S|}u \over \prod _{i\in S}\partial x_{i}}\cdot {\partial ^{n-|S|}v \over \prod _{i\not \in S}\partial x_{i}}}$

where the index S runs through all 2n subsets of {1, ..., n}, and || is the cardinality of S. For example, when n = 3,

{\displaystyle {\begin{aligned}&{\partial ^{3} \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}(uv)\\[6pt]={}&u\cdot {\partial ^{3}v \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}+{\partial u \over \partial x_{1}}\cdot {\partial ^{2}v \over \partial x_{2}\,\partial x_{3}}+{\partial u \over \partial x_{2}}\cdot {\partial ^{2}v \over \partial x_{1}\,\partial x_{3}}+{\partial u \over \partial x_{3}}\cdot {\partial ^{2}v \over \partial x_{1}\,\partial x_{2}}\\[6pt]&+{\partial ^{2}u \over \partial x_{1}\,\partial x_{2}}\cdot {\partial v \over \partial x_{3}}+{\partial ^{2}u \over \partial x_{1}\,\partial x_{3}}\cdot {\partial v \over \partial x_{2}}+{\partial ^{2}u \over \partial x_{2}\,\partial x_{3}}\cdot {\partial v \over \partial x_{1}}+{\partial ^{3}u \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}\cdot v.\end{aligned}}}

### Banach space

Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y -> Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y -> Z given by

${\displaystyle (D_{\left(x,y\right)}\,B)\left(u,v\right)=B\left(u,y\right)+B\left(x,v\right)\qquad \forall (u,v)\in X\times Y.}$

### Derivations in abstract algebra

In abstract algebra, the product rule is used to define what is called a derivation, not vice versa.

### In vector calculus

The product rule extends to scalar multiplication, dot products, and cross products of vector functions, as follows.[5]

• For scalar multiplication:
${\displaystyle (f\cdot \mathbf {g} )'=f'\cdot \mathbf {g} +f\cdot \mathbf {g} '}$
• For dot products:
${\displaystyle (\mathbf {f} \cdot \mathbf {g} )'=\mathbf {f} '\cdot \mathbf {g} +\mathbf {f} \cdot \mathbf {g} '}$
• For cross products:
${\displaystyle (\mathbf {f} \times \mathbf {g} )'=\mathbf {f} '\times \mathbf {g} +\mathbf {f} \times \mathbf {g} '}$

There are also analogues for other analogs of the derivative: if f and g are scalar fields then there is a product rule with the gradient:

${\displaystyle \nabla (f\cdot g)=\nabla f\cdot g+f\cdot \nabla g}$

## Applications

Among the applications of the product rule is a proof that

${\displaystyle {d \over dx}x^{n}=nx^{n-1}}$

when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0. The rule holds in that case because the derivative of a constant function is 0. If the rule holds for any particular exponent n, then for the next value, n + 1, we have

{\displaystyle {\begin{aligned}{d \over dx}x^{n+1}&{}={d \over dx}\left(x^{n}\cdot x\right)\\[12pt]&{}=x{d \over dx}x^{n}+x^{n}{d \over dx}x\qquad {\mbox{(the product rule is used here)}}\\[12pt]&{}=x\left(nx^{n-1}\right)+x^{n}\cdot 1\qquad {\mbox{(the induction hypothesis is used here)}}\\[12pt]&{}=(n+1)x^{n}.\end{aligned}}}

Therefore, if the proposition is true for n, it is true also for n + 1, and therefore for all natural n.

## References

1. ^ "Leibniz rule - Encyclopedia of Mathematics".
2. ^ Michelle Cirillo (August 2007). "Humanizing Calculus". The Mathematics Teacher. 101 (1): 23-27. doi:10.5951/MT.101.1.0023.
3. ^ Leibniz, G. W. (2005) [1920], The Early Mathematical Manuscripts of Leibniz (PDF), translated by J.M. Child, Dover, p. 28, footnote 58, ISBN 978-0-486-44596-0
4. ^ Micheal Hardy (January 2006). "Combinatorics of Partial Derivatives" (PDF). The Electronic Journal of Combinatorics. 13. arXiv:math/0601149. Bibcode:2006math......1149H.
5. ^ Stewart, James (2016), Calculus (8 ed.), Cengage, Section 13.2.