https://file.aiquickdraw.com/mathbot/file/1730859499490b00vzb8h.jpg

To find the first-order partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) of the function \( z = 7xy + 3x^4 y^2 \), let's go through each derivative.

### Given function:
\[ z = 7xy + 3x^4 y^2 \]

### Step 1: Calculate \(\frac{\partial z}{\partial x}\)

To find \(\frac{\partial z}{\partial x}\), we treat \(y\) as a constant and differentiate with respect to \(x\).

1. **Differentiate \(7xy\):**
   \[
   \frac{\partial}{\partial x}(7xy) = 7y
   \]

2. **Differentiate \(3x^4 y^2\):**
   \[
   \frac{\partial}{\partial x}(3x^4 y^2) = 3 \cdot 4x^3 \cdot y^2 = 12x^3 y^2
   \]

Combine these results:
\[
\frac{\partial z}{\partial x} = 7y + 12x^3 y^2
\]
This matches the provided solution.

### Step 2: Calculate \(\frac{\partial z}{\partial y}\)

Now, for \(\frac{\partial z}{\partial y}\), we treat \(x\) as a constant and differentiate with respect to \(y\).

1. **Differentiate \(7xy\):**
   \[
   \frac{\partial}{\partial y}(7xy) = 7x
   \]

2. **Differentiate \(3x^4 y^2\):**
   \[
   \frac{\partial}{\partial y}(3x^4 y^2) = 3x^4 \cdot 2y = 6x^4 y
   \]

Combine these results:
\[
\frac{\partial z}{\partial y} = 7x + 6x^4 y
\]

### Final Answers:
- \(\frac{\partial z}{\partial x} = 7y + 12x^3 y^2\)
- \(\frac{\partial z}{\partial y} = 7x + 6x^4 y\)

Would you like further details or have any questions?

### Related Questions:
1. How do you find higher-order partial derivatives of \(z\) with respect to \(x\) and \(y\)?
2. What are mixed partial derivatives, and how would you calculate \(\frac{\partial^2 z}{\partial x \partial y}\) for this function?
3. How do you interpret the meaning of partial derivatives in the context of this function?
4. What would happen if we change the power of \(x\) or \(y\) in each term? How would that affect the derivatives?
5. Can we use partial derivatives to find the critical points of \(z\)? How would that process look?

### Tip:
When differentiating terms with multiple variables, remember to treat all variables except the one you're differentiating with as constants. This helps avoid common errors in partial differentiation.

Find expressions for the first-order partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) of \( z = 7xy + 3x^4 y^2 \).

Learn how to find the first-order partial derivatives of the function z = 7xy + 3x^4 y^2 with respect to x and y. This step-by-step solution covers partial differentiation and the use of constants in multivariable calculus, making it suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation