page_title: Directional derivative
section_title: Definition
hierarchical_section_title: Directional derivative / Definition
context_page_description: In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
  The directional derivative is a special case of the Gateaux derivative.
context_section_description: The directional derivative of a scalar function 
  along a vector
  is the function  defined by the limit
  This definition is valid in a broad range of contexts, for example where the norm of a vector (and hence a unit vector) is undefined.
  If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has
  where the  on the right denotes the gradient and  is the dot product. This follows from defining a path  and using the definition of the derivative as a limit which can be calculated along this path to get:
  Intuitively, the directional derivative of f at a point x represents the rate of change of f, in the direction of v with respect to time, when moving past x.
caption: A contour plot of 
    
      
        
          f
          (
          x
          ,
          y
          )
          =
          
            x
            
              2
            
          
          +
          
            y
            
              2
            
          
        
      
      {\displaystyle f(x,y)=x^{2}+y^{2}}
    
  , showing the gradient vector in black, and the unit vector 
    
      
        
          
            u
          
        
      
      {\displaystyle \mathbf {u} }
    
   scaled by the directional derivative in the direction of 
    
      
        
          
            u
          
        
      
      {\displaystyle \mathbf {u} }
    
   in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.