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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.