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The directional derivative generalizes the partial derivative by replacing axis-aligned displacements with an arbitrary direction vector scaled by a small scalar $h$. It measures the infinitesimal output change as $h$ approaches zero, extending the concept of partial derivatives from coordinate axes to any vector direction in the input plane.

Conditions: The function is a scalar-valued function of two variables.; The partial derivative is defined along coordinate axes.; The directional derivative uses an arbitrary vector direction.

The directional derivative generalizes the partial derivative by replacing axis-aligned displacements with an arbitrary direction vector scaled by a small scalar $h$. It measures the infinitesimal output change as $h$ approaches zero, extending the concept of partial derivatives from coordinate axes to any vector direction in the input plane.

Conditions: The function is a scalar-valued function of two variables.; The partial derivative is defined along coordinate axes.; The directional derivative uses an arbitrary vector direction.