Editing Chain rule (section)

== Statement ==

The simplest form of the chain rule is for real-valued functions of one [[real number|real]] variable. It states that if ''{{mvar|g}}'' is a function that is differentiable at a point ''{{mvar|c}}'' (i.e.&nbsp;the derivative {{math|''g''′(''c'')}} exists) and ''{{mvar|f}}'' is a function that is differentiable at {{math|''g''(''c'')}}, then the composite function <math>f \circ g</math> is differentiable at ''{{mvar|c}}'', and the derivative is<ref>{{cite book|title=Mathematical analysis|author-link=Tom Apostol|first=Tom|last=Apostol|year=1974|edition=2nd|publisher=Addison Wesley|page=Theorem 5.5|no-pp=true}}</ref>
<math display="block"> (f\circ g)'(c) = f'(g(c))\cdot g'(c). </math>
The rule is sometimes abbreviated as
<math display="block">(f\circ g)' = (f'\circ g) \cdot g'.</math>

If {{math|1=''y'' = ''f''(''u'')}} and {{math|1=''u'' = ''g''(''x'')}}, then this abbreviated form is written in [[Leibniz notation]] as:
<math display="block">\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>

The points where the derivatives are evaluated may also be stated explicitly:
<math display="block">\left.\frac{dy}{dx}\right|_{x=c} = \left.\frac{dy}{du}\right|_{u = g(c)} \cdot \left.\frac{du}{dx}\right|_{x=c}.</math>

Carrying the same reasoning further, given ''{{mvar|n}}'' functions <math>f_1, \ldots, f_n\!</math> with the composite function <math>f_1 \circ ( f_2 \circ \cdots (f_{n-1} \circ f_n) )\!</math>, if each function <math>f_i\!</math> is differentiable at its immediate input, then the composite function is also differentiable by the repeated application of Chain Rule, where the derivative is (in Leibniz's notation):
<math display="block">\frac{df_1}{dx} = \frac{df_1}{df_2}\frac{df_2}{df_3}\cdots\frac{df_n}{dx}.</math>