Chain rule

In <a href="/facts/Calculus/MEmUTYoz">calculus</a>, the chain rule is a <a href="/facts/Formula/BosTcKnv">formula</a> that expresses the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of the <a href="/facts/Function_composition/r5Pvnmth">composition</a> of two <a href="/facts/Differentiable_function/jQqTgLk1">differentiable functions</a> f and g in terms of the derivatives of f and g. More precisely, if 
 
 
 
 h
 =
 f
 ∘
 g
 
 
 {\displaystyle h=f\circ g}
 
 is the function such that 
 
 
 
 h
 (
 x
 )
 =
 f
 (
 g
 (
 x
 )
 )
 
 
 {\displaystyle h(x)=f(g(x))}
 
 for every x, then the chain rule is, in <a href="/facts/Lagrange%2527s_notation/PTCbwS0Y">Lagrange's notation</a>,

h
          ′
        
        (
        x
        )
        =
        
          f
          ′
        
        (
        g
        (
        x
        )
        )
        
          g
          ′
        
        (
        x
        )
        .
      
    
    {\displaystyle h'(x)=f'(g(x))g'(x).}

or, equivalently,

h
          ′
        
        =
        (
        f
        ∘
        g
        
          )
          ′
        
        =
        (
        
          f
          ′
        
        ∘
        g
        )
        ⋅
        
          g
          ′
        
        .
      
    
    {\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}

The chain rule may also be expressed in <a href="/facts/Leibniz%2527s_notation/FLytWNf9">Leibniz's notation</a>. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are <a href="/facts/Dependent_variable/dINfzIF2">dependent variables</a>), then z depends on x as well, via the intermediate variable y. In this case, the chain rule is expressed as

d
              z
            
            
              d
              x
            
          
        
        =
        
          
            
              d
              z
            
            
              d
              y
            
          
        
        ⋅
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        ,
      
    
    {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}
  
 
and

d
                  z
                
                
                  d
                  x
                
              
            
            |
          
          
            x
          
        
        =
        
          
            
            
              
                
                  d
                  z
                
                
                  d
                  y
                
              
            
            |
          
          
            y
            (
            x
            )
          
        
        ⋅
        
          
            
            
              
                
                  d
                  y
                
                
                  d
                  x
                
              
            
            |
          
          
            x
          
        
        ,
      
    
    {\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}

for indicating at which points the derivatives have to be evaluated.
In <a href="/facts/Integral/4PeM0mJc">integration</a>, the counterpart to the chain rule is the <a href="/facts/Substitution_rule/NvXfTTaf">substitution rule</a>.

Chain rule open-in-new

Chain rule