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1. **How do higher-order derivatives help in understanding the behavior of a function?**

   Higher-order derivatives provide insights into the concavity, curvature, and rate of change of the rate of change of a function. For instance, the second derivative tells us whether a function is concave up or down, which can indicate local minima or maxima. The third derivative can describe how the concavity itself changes.

2. **What is the significance of mixed partial derivatives like \( f_{xy} \)?**

   Mixed partial derivatives measure the rate of change of one variable while another variable is also changing. They help analyze how variables interact with each other in multivariable functions. According to Clairaut's theorem, if the mixed partial derivatives are continuous, \( f_{xy} = f_{yx} \).

3. **How do you verify the equality of mixed partial derivatives, such as \( f_{xyy} = f_{yxy} \)?**

   To verify the equality of mixed partial derivatives, you differentiate the function in the specified order for each derivative. If the derivatives are equal at a given point and satisfy the conditions of Clairaut's theorem (the derivatives are continuous around that point), then they are considered equal.

4. **What would the fourth-order derivative \( f_{xyyy} \) look like for the same function?**

   Let's calculate \( f_{xyyy} \) for the function \( f(x, y) = y^{11} e^{-7x} \):

   - We already have \( f_{xyy} = -770y^9 e^{-7x} \).
   - Differentiating \( f_{xyy} \) again with respect to \( y \):
     
     \[
     f_{xyyy} = \frac{\partial}{\partial y} \left( -770y^9 e^{-7x} \right) = -770 \cdot 9y^8 e^{-7x} = -6930y^8 e^{-7x}
     \]

   So, the fourth-order derivative is:

   \[
   f_{xyyy} = -6930y^8 e^{-7x}
   \]

5. **How does \( f_{xyy} \) change if the function is altered to \( f(x, y) = y^{12} e^{-7x} \)?**

   If we modify the function to \( f(x, y) = y^{12} e^{-7x} \):

   - The first partial derivative with respect to \( x \) is:

     \[
     f_x = -7y^{12} e^{-7x}
     \]

   - Differentiating \( f_x \) with respect to \( y \):

     \[
     f_{xy} = -7 \cdot 12y^{11} e^{-7x} = -84y^{11} e^{-7x}
     \]

   - Differentiating \( f_{xy} \) again with respect to \( y \):

     \[
     f_{xyy} = -84 \cdot 11y^{10} e^{-7x} = -924y^{10} e^{-7x}
     \]

   So, \( f_{xyy} \)

This problem set covers key topics in multivariable calculus, including the calculation and interpretation of partial and higher-order derivatives. It explores the use of Clairaut’s Theorem for verifying the equality of mixed partial derivatives. Suitable for university-level calculus courses.

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