Editing Chain rule (section)

{{Short description|For derivatives of composed functions}}
{{about|composed derivatives in calculus|the probability theory concept|Chain rule (probability)|other uses}}
{{Calculus |Differential}}

In [[calculus]], the '''chain rule''' is a [[formula]] that expresses the [[derivative]] of the [[Function composition|composition]] of two [[differentiable function]]s {{mvar|f}} and {{mvar|g}} in terms of the derivatives of {{mvar|f}} and {{mvar|g}}. More precisely, if <math>h=f\circ g</math> is the function such that <math>h(x)=f(g(x))</math> for every {{mvar|x}}, then the chain rule is, in [[Lagrange's notation]],
<math display="block">h'(x) = f'(g(x)) g'(x).</math>
or, equivalently,
<math display="block">h'=(f\circ g)'=(f'\circ g)\cdot g'.</math>

The chain rule may also be expressed in [[Leibniz's notation]]. If a variable {{mvar|z}} depends on the variable {{mvar|y}}, which itself depends on the variable {{mvar|x}} (that is, {{mvar|y}} and {{mvar|z}} are [[dependent variable]]s), then {{mvar|z}} depends on {{mvar|x}} as well, via the intermediate variable {{mvar|y}}. In this case, the chain rule is expressed as
<math display="block">\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx},</math> 
and
<math display="block"> \left.\frac{dz}{dx}\right|_{x} = \left.\frac{dz}{dy}\right|_{y(x)}
\cdot \left. \frac{dy}{dx}\right|_{x} ,</math>
for indicating at which points the derivatives have to be evaluated.

In [[integral|integration]], the counterpart to the chain rule is the [[substitution rule]].