**Finding the Gradient Vector Field**In this exercise, we aim to find the gradient vector field of the function $ f(x, y) = \ln(x + 4y) $.### Function:\[ f(x, y) = \ln(x + 4y) \]### Gradient Vector Field:The gradient vector field $ \nabla f $ of a function $ f(x, y) $ is defined as:\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]### Steps:1. **Partial Derivative with respect to $ x $:**\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left[ \ln(x + 4y) \right] \]2. **Partial Derivative with respect to $ y $:**\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left[ \ln(x + 4y) \right] \]### Input Fields:There are two input fields to represent the components of the gradient vector:\[ \left( \ \ \quad , \ \ \ \right) \]Insert the solutions from the steps above into the fields to complete the problem.### Buttons:- **Add Work:** This button allows you to input additional detailed steps to show your work.---This component is designed to facilitate the understanding and calculation of gradient vectors, a fundamental concept in multivariable calculus.

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the gradient vector field of f(z, y) = In(x + 4y) %3D Work

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Finding the Gradient Vector Field**

In this exercise, we aim to find the gradient vector field of the function $ f(x, y) = \ln(x + 4y) $.

### Function:
\[ f(x, y) = \ln(x + 4y) \]

### Gradient Vector Field:
The gradient vector field $ \nabla f $ of a function $ f(x, y) $ is defined as:
\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \]

### Steps:
1. **Partial Derivative with respect to $ x $:**
\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left[ \ln(x + 4y) \right] \]

2. **Partial Derivative with respect to $ y $:**
\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left[ \ln(x + 4y) \right] \]

### Input Fields:
There are two input fields to represent the components of the gradient vector:
\[ \left( \ \ \quad , \ \ \ \right) \]

Insert the solutions from the steps above into the fields to complete the problem.

### Buttons:
- **Add Work:** This button allows you to input additional detailed steps to show your work.

---

This component is designed to facilitate the understanding and calculation of gradient vectors, a fundamental concept in multivariable calculus.

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