What would you like to understand?

To calculate the directional derivative for a specific vector, multiply each partial derivative by the corresponding component of the direction vector and sum them. For $v=[-1,2]$, this means taking -1 times the partial derivative with respect to x, plus 2 times the partial derivative with respect to y.

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

To calculate the directional derivative for a specific vector, multiply each partial derivative by the corresponding component of the direction vector and sum them. For $v=[-1,2]$, this means taking -1 times the partial derivative with respect to x, plus 2 times the partial derivative with respect to y.

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 in the direction of a general vector $w=[a,b]$ is the linear combination of the partial derivatives weighted by the components of w. The formula is a*(∂f/∂x) + b*(∂$f/y$).

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $w=[a,b]$.; The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

The directional derivative in the direction of a general vector $w=[a,b]$ is the linear combination of the partial derivatives weighted by the components of w. The formula is a*(∂f/∂x) + b*(∂$f/y$).

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $w=[a,b]$.; The partial derivatives ∂$f/x$ and ∂$f/y$ exist.

The partial derivative measures the infinitesimal output change when nudging the input strictly along the x or y axis. The directional derivative generalizes this by replacing the axis-aligned displacement with an arbitrary direction vector, scaled by a small scalar h, and taking the limit as h approaches zero.

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 partial derivative measures the infinitesimal output change when nudging the input strictly along the x or y axis. The directional derivative generalizes this by replacing the axis-aligned displacement with an arbitrary direction vector, scaled by a small scalar h, and taking the limit as h approaches zero.

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.