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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 $\mathbf{v}$ with a scalar $h$ and taking the limit as $h \to 0$, the definition captures the local slope along that specific direction. Using the full vector $\mathbf{v}$ 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 $\mathbf{v}$ with a scalar $h$ and taking the limit as $h \to 0$, the definition captures the local slope along that specific direction. Using the full vector $\mathbf{v}$ 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.