The image you uploaded contains two problems related to calculating gradients and directional derivatives for scalar fields. Let's break down each part:

Problem 1:

Given the function $f(x, y) = x^2 \ln(y)$ and the vector $\mathbf{u} = -\frac{5}{13} \hat{i} + \frac{12}{13} \hat{j}$, the tasks are:

(a) Find the gradient of $f(x, y)$:

The gradient of $f(x, y)$ is the vector of partial derivatives with respect to $x$ and $y$. This can be written as: $\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$

(b) Evaluate the gradient at the point $P(4, 1)$:

Substitute $x = 4$ and $y = 1$ into the gradient expression found in part (a).

(c) Find the rate of change of $f$ at $P$ in the direction of the vector $\mathbf{u}$:

This is done by calculating the directional derivative $D_{\mathbf{u}} f(4, 1)$, which is given by: $D_{\mathbf{u}} f(4, 1) = \nabla f(4, 1) \cdot \mathbf{u}$ where $\cdot$ denotes the dot product between the gradient and the unit vector $\mathbf{u}$.

Problem 2:

Given the function $f(x, y, z) = x^2 y z^3 - x y z$ and the point $P(2, -1, 1)$, with the vector $\mathbf{u} = \left\langle \frac{4}{5}, -\frac{3}{5} \right\rangle$, the tasks are:

(a) Find the gradient of $f(x, y, z)$:

The gradient of $f(x, y, z)$ is the vector of partial derivatives with respect to $x$, $y$, and $z$, given by: $\nabla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$

(b) Evaluate the gradient at the point $P(2, -1, 1)$:

Substitute $x = 2$, $y = -1$, and $z = 1$ into the gradient expression found in part (a).

(c) Find the rate of change of $f$ at $P$ in the direction of the vector $\mathbf{u}$:

This is similar to the previous part and involves calculating the directional derivative $D_{\mathbf{u}} f(2, -1, 1)$ using the formula: $D_{\mathbf{u}} f(2, -1, 1) = \nabla f(2, -1, 1) \cdot \mathbf{u}$

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