Question

Find the directional derivative of the function g(p, q) = p⁴ - p²q³ at the point (1, 1) in the direction of the vector v = i + 5j.

Answer 1

The directional derivative of the function
`g(p, q) = p^4 - p^2q^3 \)` at the point (1, 1) in the direction of the vector
`\( \mathbf{v} = \mathbf{i} + 5\mathbf{j} \)` is 16.

Step-by-step explanation:

The directional derivative represents the rate of change of a function in a specific direction. To compute it at the point (1, 1) in the direction of the vector
`\( \mathbf{v} = \mathbf{i} + 5\mathbf{j} \)`, start by finding the gradient of the function g(p, q). The gradient of g is
`\( \\abla g = \left((\partial g)/(\partial p), (\partial g)/(\partial q)\right) = (4p^3 - 2pq^3, -3p^2q^2) \)`.

Next, evaluate the gradient at the point (1, 1):
`\( \\abla g(1, 1) = (4 - 2, -3) = (2, -3) \).`

The directional derivative
`\( D_{\mathbf{v}}g \)` in the direction of
`\( \mathbf{v} \)` is calculated by taking the dot product of the gradient of g at the point with the unit vector in the direction of
`\( \mathbf{v} \)`:
`\( D_{\mathbf{v}}g = \\abla g(1, 1) \cdot \frac{\mathbf{v}}{\|\mathbf{v}\|} \)`.

The given vector
`\( \mathbf{v} = \mathbf{i} + 5\mathbf{j} \)` has a magnitude of
`\( √(1^2 + 5^2) = √(26) \)`. The unit vector in the direction of
`\( \mathbf{v} \)` is
`\( \frac{\mathbf{v}}{\|\mathbf{v}\|} = (1)/(√(26))\mathbf{i} + (5)/(√(26))\mathbf{j} \).`

Finally, compute the directional derivative:
`\( D_{\mathbf{v}}g = (2, -3) \cdot \left((1)/(√(26))\mathbf{i} + (5)/(√(26))\mathbf{j}\right) = (2)/(√(26)) - (15)/(√(26)) = (-13)/(√(26)) = -(13√(26))/(26) = 16 \).`