To compute the directional derivative of $f(x, y) = x^2 - y^2$ at the point $(2, 1)$ in the direction of $\vec{v} = (1, 1)$, we proceed as follows:

Step 1: Normalize the direction vector $\vec{v}$

The directional derivative requires a unit vector. The magnitude of $\vec{v} = (1, 1)$ is: $\|\vec{v}\| = \sqrt{1^2 + 1^2} = \sqrt{2}.$ The unit vector $\hat{v}$ is: $\hat{v} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right).$

Step 2: Compute the gradient of $f(x, y)$

The gradient of $f(x, y)$ is: $\nabla f(x, y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right).$ For $f(x, y) = x^2 - y^2$: $\frac{\partial f}{\partial x} = 2x, \quad \frac{\partial f}{\partial y} = -2y.$ Thus: $\nabla f(x, y) = (2x, -2y).$

Step 3: Evaluate $\nabla f(x, y)$ at $(2, 1)$

At $(2, 1)$, the gradient is: $\nabla f(2, 1) = (2(2), -2(1)) = (4, -2).$

Step 4: Compute the directional derivative

The directional derivative in the direction of $\hat{v}$ is given by: $D_{\hat{v}}f(2, 1) = \nabla f(2, 1) \cdot \hat{v}.$ Substitute $\nabla f(2, 1) = (4, -2)$ and $\hat{v} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$: $D_{\hat{v}}f(2, 1) = (4, -2) \cdot \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right).$ Compute the dot product: $D_{\hat{v}}f(2, 1) = 4\left(\frac{1}{\sqrt{2}}\right) + (-2)\left(\frac{1}{\sqrt{2}}\right) = \frac{4}{\sqrt{2}} - \frac{2}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}.$

Final Answer:

The directional derivative is: $D_{\hat{v}}f(2, 1) = \sqrt{2}.$

Let me know if you'd like further clarification or additional examples! Here are five related questions:

1. What is the geometric interpretation of the gradient in a function?
2. How do you compute the magnitude of a vector in $n$-dimensional space?
3. What happens to the directional derivative if the direction vector is not normalized?
4. Can the directional derivative be negative? If so, what does it signify?
5. How does the directional derivative change at different points of the function?

Tip: Always normalize the direction vector when computing directional derivatives!