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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.

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.

For a differentiable function $f(x,y)$, the directional derivative along the vector $v=[-1,2]$ is calculated by multiplying the partial derivative with respect to $x$ by $-1$ and the partial derivative with respect to $y$ by $2$, then summing these products. This yields the expression $-\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$, assuming the partial derivatives exist.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $v=[-1,2]$.; The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

For a differentiable function $f(x,y)$, the directional derivative along the vector $v=[-1,2]$ is calculated by multiplying the partial derivative with respect to $x$ by $-1$ and the partial derivative with respect to $y$ by $2$, then summing these products. This yields the expression $-\frac{\partial f}{\partial x} + 2\frac{\partial f}{\partial y}$, assuming the partial derivatives exist.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $v=[-1,2]$.; The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

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.