Activity 11.8.2

In this activity, we will use equation to calculate and interpret directional derivatives for the function $f(x,y) = 3xy-x^2y^3$.

1. Calculate $f_x(x,y)$ and $f_y(x,y)$.

2. Use equation to determine $D_{\vi} f(x,y)$ and $D_{\vj} f(x,y)$. Write a couple of sentences to describe what familiar functions $D_{\vi} f$ and $D_{\vj} f$ are.

3. Use equation to find the derivative of $f$ in the direction of the vector $\vv = \langle 2, 3 \rangle$ at the point $(1,-1)$.

4. Find the derivative of $f$ in the direction of the vector $\vv = \langle 4, 6 \rangle$ at the point $(1,-1)$.

5. Use equation to find the derivative of $f$ in the direction of the vector $\vv = \langle -2, -3 \rangle$ at the point $(1,-1)$. Write a couple of sentences to explain why this result is different from your answer to the previous two tasks, even though the direction vectors are parallel.