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The directional derivative of a scalar function of two variables at a point represents the rate of change of the function's output when the input is nudged infinitesimally in a chosen direction. Geometrically, it is the slope of the tangent line to the curve formed by intersecting the surface with a vertical plane passing through the point and parallel to the direction vector.

Conditions: The function is a scalar-valued function of two variables.; The direction is specified by a vector in the input plane.; The step size approaches zero.

The directional derivative of a scalar function of two variables at a point represents the rate of change of the function's output when the input is nudged infinitesimally in a chosen direction. Geometrically, it is the slope of the tangent line to the curve formed by intersecting the surface with a vertical plane passing through the point and parallel to the direction vector.

Conditions: The function is a scalar-valued function of two variables.; The direction is specified by a vector in the input plane.; The step size approaches zero.

For a differentiable function $f(x,y)$, the directional derivative in the direction of vector $\mathbf{w} = [a,b]$ is given by the dot product $\mathbf{w} \cdot \nabla f$, where $\nabla f = [\partial f/\partial x, \partial f/\partial y]$. This formulation holds without requiring normalization of $\mathbf{w}$.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $w=[a,b]$.; The gradient ∇f = [∂f/∂x, ∂f/∂y] exists.

For a differentiable function $f(x,y)$, the directional derivative in the direction of vector $\mathbf{w} = [a,b]$ is given by the dot product $\mathbf{w} \cdot \nabla f$, where $\nabla f = [\partial f/\partial x, \partial f/\partial y]$. This formulation holds without requiring normalization of $\mathbf{w}$.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $w=[a,b]$.; The gradient ∇f = [∂f/∂x, ∂f/∂y] exists.