Differential of a function

In <a href="/facts/Calculus/MEmUTYoz">calculus</a>, the differential represents the <a href="/facts/Principal_part/QN7kHcey">principal part</a> of the change in a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 y
 =
 f
 (
 x
 )
 
 
 {\displaystyle y=f(x)}
 
 with respect to changes in the independent variable. The differential 
 
 
 
 d
 y
 
 
 {\displaystyle dy}
 
 is defined by

d
        y
        =
        
          f
          ′
        
        (
        x
        )
        
        d
        x
        ,
      
    
    {\displaystyle dy=f'(x)\,dx,}

where 
 
 
 
 
 f
 ′
 
 (
 x
 )
 
 
 {\displaystyle f'(x)}
 
 is the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of f with respect to 
 
 
 
 x
 
 
 {\displaystyle x}
 
, and 
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
 is an additional real <a href="/facts/Variable_(mathematics)/VbQlrvKz">variable</a> (so that 
 
 
 
 d
 y
 
 
 {\displaystyle dy}
 
 is a function of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
). The notation is such that the equation

 
 
 
 d
 y
 =
 
 
 
 d
 y
 
 
 d
 x
 
 
 
 
 d
 x
 
 
 {\displaystyle dy={\frac {dy}{dx}}\,dx}

holds, where the derivative is represented in the <a href="/facts/Leibniz_notation/FLytWNf9">Leibniz notation</a> 
 
 
 
 d
 y
 
 /
 
 d
 x
 
 
 {\displaystyle dy/dx}
 
, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

 
 
 
 d
 f
 (
 x
 )
 =
 
 f
 ′
 
 (
 x
 )
 
 d
 x
 .
 
 
 {\displaystyle df(x)=f'(x)\,dx.}

The precise meaning of the variables 
 
 
 
 d
 y
 
 
 {\displaystyle dy}
 
 and 
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
 depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular <a href="/facts/Differential_form/T9gyHtMv">differential form</a>, or analytical significance if the differential is regarded as a <a href="/facts/Linear_approximation/SDBAneFx">linear approximation</a> to the increment of a function. Traditionally, the variables 
 
 
 
 d
 x
 
 
 {\displaystyle dx}
 
 and 
 
 
 
 d
 y
 
 
 {\displaystyle dy}
 
 are considered to be very small (<a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a>), and this interpretation is made rigorous in <a href="/facts/Non-standard_analysis/lEpNpWMB">non-standard analysis</a>.

Differential of a function open-in-new

Differential of a function