The directional derivative is defined as an instantaneous rate of change, requiring an infinitesimal step rather than a finite one. By scaling the direction vector v with a scalar h and taking the limit as h→0, the definition captures the local slope along that specific direction. Using the full vector v alone would represent a finite displacement, which fails to describe the derivative's nature as a limit of ratios over vanishingly small intervals.
Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.
The directional derivative is defined as an instantaneous rate of change, requiring an infinitesimal step rather than a finite one. By scaling the direction vector v with a scalar h and taking the limit as h→0, the definition captures the local slope along that specific direction. Using the full vector v alone would represent a finite displacement, which fails to describe the derivative's nature as a limit of ratios over vanishingly small intervals.
Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.
The directional derivative formula a*(∂f/∂x) + b*(∂f/∂y) structurally matches the definition of a dot product between the vector [a,b] and the vector [∂f/∂x, ∂f/∂y]. Since the second vector is defined as the gradient ∇f, the expression simplifies to w·∇f.
Conditions: The function f(x,y) is differentiable.; The direction vector is w=[a,b].; The gradient ∇f = [∂f/x, ∂f/y] exists.
The directional derivative formula a*(∂f/∂x) + b*(∂f/∂y) structurally matches the definition of a dot product between the vector [a,b] and the vector [∂f/∂x, ∂f/∂y]. Since the second vector is defined as the gradient ∇f, the expression simplifies to w·∇f.
Conditions: The function f(x,y) is differentiable.; The direction vector is w=[a,b].; The gradient ∇f = [∂f/x, ∂f/y] exists.