Formula for the derivative of a product
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
( u ⋅ v ) ′ = u ′ ⋅ v + u ⋅ v ′ {\displaystyle (u\cdot v)'=u'\cdot v+u\cdot v'} or in
Leibniz's notation as
d d x ( u ⋅ v ) = d u d x ⋅ v + u ⋅ d v d x . {\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 [ edit ] 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
d ( u ⋅ v ) = ( u + d u ) ⋅ ( v + d v ) − u ⋅ v = u ⋅ d v + v ⋅ d u + d u ⋅ d v . {\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
d ( u ⋅ v ) = v ⋅ d u + u ⋅ d v {\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
d d x ( u ⋅ v ) = v ⋅ d u d x + u ⋅ d v d x {\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
( u ⋅ v ) ′ = v ⋅ u ′ + u ⋅ v ′ . {\displaystyle (u\cdot v)'=v\cdot u'+u\cdot v'.} Examples [ edit ] Suppose we want to differentiate f ( x ) = x 2 sin ( x ) . {\displaystyle f(x)=x^{2}{\text{sin}}(x).} By using the product rule, one gets the derivative f ′ ( x ) = 2 x ⋅ sin ( x ) + x 2 cos ( x ) {\displaystyle f'(x)=2x\cdot {\text{sin}}(x)+x^{2}{\text{cos}}(x)} (since the derivative of x 2 {\displaystyle x^{2}} is 2 x , {\displaystyle 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 ) {\displaystyle f(x)} is a differentiable function, then c ⋅ f ( x ) {\displaystyle c\cdot f(x)} is also differentiable, and its derivative is ( c f ) ′ ( x ) = c ⋅ f ′ ( x ) . {\displaystyle (cf)'(x)=c\cdot f'(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.) Limit definition of derivative [ edit ] 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, f ( x ) g ( x + Δ x ) − f ( x ) g ( x + Δ x ) {\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.
h ′ ( x ) = lim Δ x → 0 h ( x + Δ x ) − h ( x ) Δ x = lim Δ x → 0 f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = lim Δ x → 0 f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x + Δ x ) + f ( x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = lim Δ x → 0 [ f ( x + Δ x ) − f ( x ) ] ⋅ g ( x + Δ x ) + f ( x ) ⋅ [ g ( x + Δ x ) − g ( x ) ] Δ x = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x ⋅ lim Δ x → 0 g ( x + Δ x ) ⏟ See the note below. + lim Δ x → 0 f ( x ) ⋅ lim Δ x → 0 g ( x + Δ x ) − g ( x ) Δ x = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) . {\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
lim Δ x → 0 g ( x + Δ x ) = g ( x ) {\displaystyle \lim _{\Delta x\to 0}g(x+\Delta x)=g(x)} follows from the fact that differentiable functions are continuous.
Linear approximations [ edit ] By definition, if f , g : R → R {\displaystyle f,g:\mathbb {R} \to \mathbb {R} } are differentiable at x {\displaystyle x} , then we can write linear approximations :
f ( x + h ) = f ( x ) + f ′ ( x ) h + ε 1 ( h ) {\displaystyle f(x+h)=f(x)+f'(x)h+\varepsilon _{1}(h)} and
g ( x + h ) = g ( x ) + g ′ ( x ) h + ε 2 ( h ) , {\displaystyle g(x+h)=g(x)+g'(x)h+\varepsilon _{2}(h),} where the error terms are small with respect to
h : that is,
lim h → 0 ε 1 ( h ) h = lim h → 0 ε 2 ( h ) h = 0 , {\textstyle \lim _{h\to 0}{\frac {\varepsilon _{1}(h)}{h}}=\lim _{h\to 0}{\frac {\varepsilon _{2}(h)}{h}}=0,} also written ε 1 , ε 2 ∼ o ( h ) {\displaystyle \varepsilon _{1},\varepsilon _{2}\sim o(h)} . Then:
f ( x + h ) g ( x + h ) − f ( x ) g ( x ) = ( f ( x ) + f ′ ( x ) h + ε 1 ( h ) ) ( g ( x ) + g ′ ( x ) h + ε 2 ( h ) ) − f ( x ) g ( x ) = f ( x ) g ( x ) + f ′ ( x ) g ( x ) h + f ( x ) g ′ ( x ) h − f ( x ) g ( x ) + error terms = f ′ ( x ) g ( x ) h + f ( x ) g ′ ( x ) h + o ( h ) . {\displaystyle {\begin{aligned}f(x+h)g(x+h)-f(x)g(x)&=(f(x)+f'(x)h+\varepsilon _{1}(h))(g(x)+g'(x)h+\varepsilon _{2}(h))-f(x)g(x)\\[.5em]&=f(x)g(x)+f'(x)g(x)h+f(x)g'(x)h-f(x)g(x)+{\text{error terms}}\\[.5em]&=f'(x)g(x)h+f(x)g'(x)h+o(h).\end{aligned}}} The "error terms" consist of items such as
f ( x ) ε 2 ( h ) , f ′ ( x ) g ′ ( x ) h 2 {\displaystyle f(x)\varepsilon _{2}(h),f'(x)g'(x)h^{2}} and
h f ′ ( x ) ε 1 ( h ) {\displaystyle hf'(x)\varepsilon _{1}(h)} which are easily seen to have magnitude
o ( h ) . {\displaystyle o(h).} Dividing by
h {\displaystyle h} and taking the limit
h → 0 {\displaystyle h\to 0} gives the result.
Quarter squares [ edit ] This proof uses the chain rule and the quarter square function q ( x ) = 1 4 x 2 {\displaystyle q(x)={\tfrac {1}{4}}x^{2}} with derivative q ′ ( x ) = 1 2 x {\displaystyle q'(x)={\tfrac {1}{2}}x} . We have:
u v = q ( u + v ) − q ( u − v ) , {\displaystyle uv=q(u+v)-q(u-v),} and differentiating both sides gives:
f ′ = q ′ ( u + v ) ( u ′ + v ′ ) − q ′ ( u − v ) ( u ′ − v ′ ) = ( 1 2 ( u + v ) ( u ′ + v ′ ) ) − ( 1 2 ( u − v ) ( u ′ − v ′ ) ) = 1 2 ( u u ′ + v u ′ + u v ′ + v v ′ ) − 1 2 ( u u ′ − v u ′ − u v ′ + v v ′ ) = v u ′ + u v ′ . {\displaystyle {\begin{aligned}f'&=q'(u+v)(u'+v')-q'(u-v)(u'-v')\\[4pt]&=\left({\tfrac {1}{2}}(u+v)(u'+v')\right)-\left({\tfrac {1}{2}}(u-v)(u'-v')\right)\\[4pt]&={\tfrac {1}{2}}(uu'+vu'+uv'+vv')-{\tfrac {1}{2}}(uu'-vu'-uv'+vv')\\[4pt]&=vu'+uv'.\end{aligned}}} Multivariable chain rule [ edit ] The product rule can be considered a special case of the chain rule for several variables, applied to the multiplication function m ( u , v ) = u v {\displaystyle m(u,v)=uv} :
d ( u v ) d x = ∂ ( u v ) ∂ u d u d x + ∂ ( u v ) ∂ v d v d x = v d u d x + u d v d x . {\displaystyle {d(uv) \over dx}={\frac {\partial (uv)}{\partial u}}{\frac {du}{dx}}+{\frac {\partial (uv)}{\partial v}}{\frac {dv}{dx}}=v{\frac {du}{dx}}+u{\frac {dv}{dx}}.} Non-standard analysis [ edit ] 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
d ( u v ) d x = st ( ( u + d u ) ( v + d v ) − u v d x ) = st ( u v + u ⋅ d v + v ⋅ d u + d u ⋅ d v − u v d x ) = st ( u ⋅ d v + v ⋅ d u + d u ⋅ d v d x ) = st ( u d v d x + ( v + d v ) d u d x ) = u d v d x + v d u d x . {\displaystyle {\begin{aligned}{\frac {d(uv)}{dx}}&=\operatorname {st} \left({\frac {(u+du)(v+dv)-uv}{dx}}\right)\\&=\operatorname {st} \left({\frac {uv+u\cdot dv+v\cdot du+du\cdot dv-uv}{dx}}\right)\\&=\operatorname {st} \left({\frac {u\cdot dv+v\cdot du+du\cdot dv}{dx}}\right)\\&=\operatorname {st} \left(u{\frac {dv}{dx}}+(v+dv){\frac {du}{dx}}\right)\\&=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 [ edit ] In the context of Lawvere's approach to infinitesimals, let d x {\displaystyle dx} be a nilsquare infinitesimal. Then d u = u ′ d x {\displaystyle du=u'\ dx} and d v = v ′ d x {\displaystyle dv=v'\ dx} , so that
d ( u v ) = ( u + d u ) ( v + d v ) − u v = u v + u ⋅ d v + v ⋅ d u + d u ⋅ d v − u v = u ⋅ d v + v ⋅ d u + d u ⋅ d v = u ⋅ d v + v ⋅ d u {\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
d u d v = u ′ v ′ ( d x ) 2 = 0. {\displaystyle du\,dv=u'v'(dx)^{2}=0.} Dividing by
d x {\displaystyle dx} then gives
d ( u v ) d x = u d v d x + v d u d x {\displaystyle {\frac {d(uv)}{dx}}=u{\frac {dv}{dx}}+v{\frac {du}{dx}}} or
( u v ) ′ = u ⋅ v ′ + v ⋅ u ′ {\displaystyle (uv)'=u\cdot v'+v\cdot u'} .
Logarithmic differentiation [ edit ] Let h ( x ) = f ( x ) g ( x ) {\displaystyle h(x)=f(x)g(x)} . Taking the absolute value of each function and the natural log of both sides of the equation,
ln | h ( x ) | = ln | f ( x ) g ( x ) | {\displaystyle \ln |h(x)|=\ln |f(x)g(x)|} Applying properties of the absolute value and logarithms,
ln | h ( x ) | = ln | f ( x ) | + ln | g ( x ) | {\displaystyle \ln |h(x)|=\ln |f(x)|+\ln |g(x)|} Taking the
logarithmic derivative of both sides and then solving for
h ′ ( x ) {\displaystyle h'(x)} :
h ′ ( x ) h ( x ) = f ′ ( x ) f ( x ) + g ′ ( x ) g ( x ) {\displaystyle {\frac {h'(x)}{h(x)}}={\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}} Solving for
h ′ ( x ) {\displaystyle h'(x)} and substituting back
f ( x ) g ( x ) {\displaystyle f(x)g(x)} for
h ( x ) {\displaystyle h(x)} gives:
h ′ ( x ) = h ( x ) ( f ′ ( x ) f ( x ) + g ′ ( x ) g ( x ) ) = f ( x ) g ( x ) ( f ′ ( x ) f ( x ) + g ′ ( x ) g ( x ) ) = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) . {\displaystyle {\begin{aligned}h'(x)&=h(x)\left({\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\right)\\&=f(x)g(x)\left({\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\right)\\&=f'(x)g(x)+f(x)g'(x).\end{aligned}}} Note: Taking the absolute value of the functions is necessary for the
logarithmic differentiation of functions that may have negative values, as logarithms are only
real-valued for positive arguments. This works because
d d x ( ln | u | ) = u ′ u {\displaystyle {\tfrac {d}{dx}}(\ln |u|)={\tfrac {u'}{u}}} , which justifies taking the absolute value of the functions for logarithmic differentiation.
Generalizations [ edit ] Product of more than two factors [ edit ] The product rule can be generalized to products of more than two factors. For example, for three factors we have
d ( u v w ) d x = d u d x v w + u d v d x w + u v d w d x . {\displaystyle {\frac {d(uvw)}{dx}}={\frac {du}{dx}}vw+u{\frac {dv}{dx}}w+uv{\frac {dw}{dx}}.} For a collection of functions
f 1 , … , f k {\displaystyle f_{1},\dots ,f_{k}} , we have
d d x [ ∏ i = 1 k f i ( x ) ] = ∑ i = 1 k ( ( d d x f i ( x ) ) ∏ j = 1 , j ≠ i k f j ( x ) ) = ( ∏ i = 1 k f i ( x ) ) ( ∑ i = 1 k f i ′ ( x ) f i ( x ) ) . {\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
Logder ( f ) = f ′ f . {\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
Logder ( f 1 ⋯ f k ) = ∑ i = 1 k Logder ( f i ) . {\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
f i . {\displaystyle f_{i}.} Higher derivatives [ edit ] It can also be generalized to the general Leibniz rule for the n th derivative of a product of two factors, by symbolically expanding according to the binomial theorem :
d n ( u v ) = ∑ k = 0 n ( n k ) ⋅ d ( n − k ) ( u ) ⋅ d ( k ) ( v ) . {\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:
( u v ) ( n ) ( x ) = ∑ k = 0 n ( n k ) ⋅ u ( n − k ) ( x ) ⋅ v ( k ) ( x ) . {\displaystyle (uv)^{(n)}(x)=\sum _{k=0}^{n}{n \choose k}\cdot u^{(n-k)}(x)\cdot v^{(k)}(x).} Furthermore, for the n th derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients :
( ∏ i = 1 k f i ) ( n ) = ∑ j 1 + j 2 + ⋯ + j k = n ( n j 1 , j 2 , … , j k ) ∏ i = 1 k f i ( j i ) . {\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 [ edit ] For partial derivatives , we have[4]
∂ n ∂ x 1 ⋯ ∂ x n ( u v ) = ∑ S ∂ | S | u ∏ i ∈ S ∂ x i ⋅ ∂ n − | S | v ∏ i ∉ S ∂ x i {\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
|S | is the
cardinality of
S . For example, when
n = 3,
∂ 3 ∂ x 1 ∂ x 2 ∂ x 3 ( u v ) = u ⋅ ∂ 3 v ∂ x 1 ∂ x 2 ∂ x 3 + ∂ u ∂ x 1 ⋅ ∂ 2 v ∂ x 2 ∂ x 3 + ∂ u ∂ x 2 ⋅ ∂ 2 v ∂ x 1 ∂ x 3 + ∂ u ∂ x 3 ⋅ ∂ 2 v ∂ x 1 ∂ x 2 + ∂ 2 u ∂ x 1 ∂ x 2 ⋅ ∂ v ∂ x 3 + ∂ 2 u ∂ x 1 ∂ x 3 ⋅ ∂ v ∂ x 2 + ∂ 2 u ∂ x 2 ∂ x 3 ⋅ ∂ v ∂ x 1 + ∂ 3 u ∂ x 1 ∂ x 2 ∂ x 3 ⋅ v . {\displaystyle {\begin{aligned}&{\partial ^{3} \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}(uv)\\[1ex]={}&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}}\\[1ex]&+{\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.\\[-3ex]&\end{aligned}}} Banach space [ edit ] 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
( D ( x , y ) B ) ( u , v ) = B ( u , y ) + B ( x , v ) ∀ ( u , v ) ∈ X × Y . {\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.} This result can be extended[5] to more general topological vector spaces.
In vector calculus [ edit ] The product rule extends to various product operations of vector functions on R n {\displaystyle \mathbb {R} ^{n}} :[6]
For scalar multiplication : ( f ⋅ g ) ′ = f ′ ⋅ g + f ⋅ g ′ {\displaystyle (f\cdot \mathbf {g} )'=f'\cdot \mathbf {g} +f\cdot \mathbf {g} '} For dot product : ( f ⋅ g ) ′ = f ′ ⋅ g + f ⋅ g ′ {\displaystyle (\mathbf {f} \cdot \mathbf {g} )'=\mathbf {f} '\cdot \mathbf {g} +\mathbf {f} \cdot \mathbf {g} '} For cross product of vector functions on R 3 {\displaystyle \mathbb {R} ^{3}} : ( f × g ) ′ = f ′ × g + f × g ′ {\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 :
∇ ( f ⋅ g ) = ∇ f ⋅ g + f ⋅ ∇ g {\displaystyle \nabla (f\cdot g)=\nabla f\cdot g+f\cdot \nabla g} Such a rule will hold for any continuous bilinear product operation. Let B : X × Y → Z be a continuous bilinear map between vector spaces, and let f and g be differentiable functions into X and Y , respectively. The only properties of multiplication used in the proof using the limit definition of derivative is that multiplication is continuous and bilinear. So for any continuous bilinear operation,
H ( f , g ) ′ = H ( f ′ , g ) + H ( f , g ′ ) . {\displaystyle H(f,g)'=H(f',g)+H(f,g').} This is also a special case of the product rule for bilinear maps in
Banach space .
Derivations in abstract algebra and differential geometry [ edit ] In abstract algebra , the product rule is the defining property of a derivation . In this terminology, the product rule states that the derivative operator is a derivation on functions.
In differential geometry , a tangent vector to a manifold M at a point p may be defined abstractly as an operator on real-valued functions which behaves like a directional derivative at p : that is, a linear functional v which is a derivation,
v ( f g ) = v ( f ) g ( p ) + f ( p ) v ( g ) . {\displaystyle v(fg)=v(f)\,g(p)+f(p)\,v(g).} Generalizing (and dualizing) the formulas of vector calculus to an
n -dimensional manifold
M, one may take
differential forms of degrees
k and
l , denoted
α ∈ Ω k ( M ) , β ∈ Ω ℓ ( M ) {\displaystyle \alpha \in \Omega ^{k}(M),\beta \in \Omega ^{\ell }(M)} , with the wedge or
exterior product operation
α ∧ β ∈ Ω k + ℓ ( M ) {\displaystyle \alpha \wedge \beta \in \Omega ^{k+\ell }(M)} , as well as the
exterior derivative d : Ω m ( M ) → Ω m + 1 ( M ) {\displaystyle d:\Omega ^{m}(M)\to \Omega ^{m+1}(M)} . Then one has the
graded Leibniz rule :
d ( α ∧ β ) = d α ∧ β + ( − 1 ) k α ∧ d β . {\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta .} Applications [ edit ] Among the applications of the product rule is a proof that
d d x x n = n x n − 1 {\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
x n is constant and
nx n − 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
d x n + 1 d x = d d x ( x n ⋅ x ) = x d d x x n + x n d d x x (the product rule is used here) = x ( n x n − 1 ) + x n ⋅ 1 (the induction hypothesis is used here) = ( n + 1 ) x n . {\displaystyle {\begin{aligned}{\frac {dx^{n+1}}{dx}}&{}={\frac {d}{dx}}\left(x^{n}\cdot x\right)\\[1ex]&{}=x{\frac {d}{dx}}x^{n}+x^{n}{\frac {d}{dx}}x&{\text{(the product rule is used here)}}\\[1ex]&{}=x\left(nx^{n-1}\right)+x^{n}\cdot 1&{\text{(the induction hypothesis is used here)}}\\[1ex]&{}=\left(n+1\right)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 .
See also [ edit ] References [ edit ] ^ "Leibniz rule – Encyclopedia of Mathematics" . ^ Michelle Cirillo (August 2007). "Humanizing Calculus" . The Mathematics Teacher . 101 (1): 23–27. doi :10.5951/MT.101.1.0023 . ^ 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 ^ Micheal Hardy (January 2006). "Combinatorics of Partial Derivatives" (PDF) . The Electronic Journal of Combinatorics . 13 . arXiv :math/0601149 . Bibcode :2006math......1149H . ^ Kreigl, Andreas; Michor, Peter (1997). The Convenient Setting of Global Analysis (PDF) . American Mathematical Society. p. 59. ISBN 0-8218-0780-3 . ^ Stewart, James (2016), Calculus (8 ed.), Cengage , Section 13.2.