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Let's go through each of the partial derivatives for the function \( f(x, y) = 8x^3 y^2 + 7y^5 \).

### 1. First Partial Derivatives

**(a) \( f_x \):**  
Differentiate \( f(x, y) = 8x^3 y^2 + 7y^5 \) with respect to \( x \), treating \( y \) as a constant.

\[
f_x = \frac{\partial}{\partial x} (8x^3 y^2 + 7y^5) = 8 \cdot 3x^2 \cdot y^2 = 24x^2 y^2
\]

So, \( f_x = 24x^2 y^2 \).

**(b) \( f_y \):**  
Differentiate \( f(x, y) = 8x^3 y^2 + 7y^5 \) with respect to \( y \), treating \( x \) as a constant.

\[
f_y = \frac{\partial}{\partial y} (8x^3 y^2 + 7y^5) = 8x^3 \cdot 2y + 7 \cdot 5y^4 = 16x^3 y + 35y^4
\]

So, \( f_y = 16x^3 y + 35y^4 \).

### 2. Second Partial Derivatives

**(c) \( f_{xx} \):**  
Differentiate \( f_x = 24x^2 y^2 \) with respect to \( x \) again.

\[
f_{xx} = \frac{\partial}{\partial x} (24x^2 y^2) = 24 \cdot 2x \cdot y^2 = 48x y^2
\]

So, \( f_{xx} = 48x y^2 \).

**(d) \( f_{yy} \):**  
Differentiate \( f_y = 16x^3 y + 35y^4 \) with respect to \( y \) again.

\[
f_{yy} = \frac{\partial}{\partial y} (16x^3 y + 35y^4) = 16x^3 + 35 \cdot 4y^3 = 16x^3 + 140y^3
\]

So, \( f_{yy} = 16x^3 + 140y^3 \).

**(e) \( f_{xy} \):**  
Differentiate \( f_x = 24x^2 y^2 \) with respect to \( y \).

\[
f_{xy} = \frac{\partial}{\partial y} (24x^2 y^2) = 24x^2 \cdot 2y = 48x^2 y
\]

So, \( f_{xy} = 48x^2 y \).

### Summary of Results

- \( f_x = 24x^2 y^2 \)
- \( f_y = 16x^3 y + 35y^4 \)
- \( f_{xx} = 48x y^2 \)
- \( f_{yy} = 16x^3 + 140y^3 \)
- \( f_{xy} = 48x^2 y \)

Would you like further details or explanations for any of these steps?

Here are five follow-up questions related to partial derivatives and their applications:

1. What is the geometric interpretation of the first partial derivatives \( f_x \) and \( f_y \)?
2. How are second partial derivatives used to determine the concavity or convexity of a function?
3. What does the mixed partial derivative \( f_{xy} \) indicate about the function's behavior?
4. How would these partial derivatives change if we introduced an additional term in \( f(x, y) \)?
5. Can you use these partial derivatives to determine critical points of \( f(x, y) \)?

**Tip:** Mixed partial derivatives often provide insights into how changes in one variable affect the rate of change in another.

Consider the function f(x, y) = 8x^3 y^2 + 7y^5. Find the first and second partial derivatives (f_x, f_y, f_xx, f_yy, f_xy).

Learn how to find the first and second partial derivatives of the function f(x, y) = 8x^3 y^2 + 7y^5, covering f_x, f_y, f_xx, f_yy, and f_xy. This solution provides detailed steps for each derivative, focusing on partial derivatives in multivariable calculus. Suitable for undergraduate students.

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