Here's a simple exercise on directional derivatives:

---

Exercise:

Let $f(x, y) = x^2 + 3y^2$.

1. Find the directional derivative of $f(x, y)$ at the point $P(1, 2)$ in the direction of the vector $\mathbf{v} = \langle 3, 4 \rangle$.

---

Solution Steps Outline:

1. Normalize the direction vector $\mathbf{v} = \langle 3, 4 \rangle$.
2. Compute the gradient of $f(x, y)$, $\nabla f(x, y)$.
3. Evaluate $\nabla f(x, y)$ at $P(1, 2)$.
4. Compute the dot product of $\nabla f(P)$ with the normalized vector to get the directional derivative.

Would you like me to solve this step-by-step or help with another similar exercise? 😊

---

5 Related Questions:

1. How do you compute the gradient of a multivariable function?
2. What is the significance of normalizing a direction vector in directional derivatives?
3. Can directional derivatives be negative? Why or why not?
4. How is the directional derivative related to partial derivatives?
5. What is the maximum rate of change of a function, and how is it connected to the gradient?

---

Tip:

The gradient $\nabla f$ points in the direction of the greatest rate of increase of $f(x, y)$.