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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.