Find f Subscript xx​, f Subscript xy​, Subscript  f Subscript yx​, and f Subscript yy for the following function.​ (Remember, f Subscript yx means to differentiate with respect to y and then with respect to​ x.)
​f(x, ​y)equals10yln x

Given the function \( f(x, y) = 10y \ln(x) \), we need to find the following partial derivatives:

1. \( f_{xx} \) (the second partial derivative of \( f \) with respect to \( x \)),
2. \( f_{xy} \) (the partial derivative of \( f \) first with respect to \( x \) and then with respect to \( y \)),
3. \( f_{yx} \) (the partial derivative of \( f \) first with respect to \( y \) and then with respect to \( x \)), and
4. \( f_{yy} \) (the second partial derivative of \( f \) with respect to \( y \)).

Let's start with the first-order partial derivatives:

### 1. First-Order Partial Derivatives

#### \( f_x \) (the partial derivative of \( f \) with respect to \( x \)):
\[
f_x = \frac{\partial}{\partial x} \left( 10y \ln(x) \right)
\]
Since \( 10y \) is treated as a constant with respect to \( x \), we have:
\[
f_x = 10y \cdot \frac{1}{x} = \frac{10y}{x}
\]

#### \( f_y \) (the partial derivative of \( f \) with respect to \( y \)):
\[
f_y = \frac{\partial}{\partial y} \left( 10y \ln(x) \right)
\]
Since \( \ln(x) \) is treated as a constant with respect to \( y \), we have:
\[
f_y = 10 \ln(x)
\]

### 2. Second-Order Partial Derivatives

#### \( f_{xx} \) (the second partial derivative of \( f \) with respect to \( x \)):
\[
f_{xx} = \frac{\partial}{\partial x} \left( \frac{10y}{x} \right)
\]
Taking the derivative of \( \frac{10y}{x} \) with respect to \( x \):
\[
f_{xx} = -\frac{10y}{x^2}
\]

#### \( f_{xy} \) (the partial derivative of \( f_x \) with respect to \( y \)):
\[
f_{xy} = \frac{\partial}{\partial y} \left( \frac{10y}{x} \right)
\]
Taking the derivative of \( \frac{10y}{x} \) with respect to \( y \):
\[
f_{xy} = \frac{10}{x}
\]

#### \( f_{yx} \) (the partial derivative of \( f_y \) with respect to \( x \)):
\[
f_{yx} = \frac{\partial}{\partial x} \left( 10 \ln(x) \right)
\]
Taking the derivative of \( 10 \ln(x) \) with respect to \( x \):
\[
f_{yx} = \frac{10}{x}
\]
(Note that \( f_{xy} = f_{yx} \), which is expected due to Clairaut's theorem for mixed partial derivatives.)

#### \( f_{yy} \) (the second partial derivative of \( f \) with respect to \( y \)):
\[
f_{yy} = \frac{\partial}{\partial y} \left( 10 \ln(x) \right)
\]
Since \( 10 \ln(x) \) is independent of \( y \), we have:
\[
f_{yy} = 0
\]

### Summary of Results

- \( f_{xx} = -\frac{10y}{x^2} \)
- \( f_{xy} = \frac{10}{x} \)
- \( f_{yx} = \frac{10}{x} \)
- \( f_{yy} = 0 \)

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How do we use the second-order partial derivatives in determining the concavity of a function?
2. What is the significance of mixed partial derivatives being equal (i.e., \( f_{xy} = f_{yx} \))?
3. How would the partial derivatives change if the function were \( f(x, y) = 10x \ln(y) \) instead?
4. Can you find the critical points of the function \( f(x, y) = 10y \ln(x) \)?
5. How do partial derivatives relate to the gradient of a function in multivariable calculus?

**Tip:** In multivariable calculus, the equality \( f_{xy} = f_{yx} \) is guaranteed under certain conditions (such as continuity of second-order derivatives) and is known as Schwarz's theorem or Clairaut's theorem.

Learn how to find second-order partial derivatives for the function f(x, y) = 10y ln(x). Includes detailed calculations for f_xx, f_xy, f_yx, and f_yy. Suitable for advanced undergraduate students studying multivariable calculus.

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