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.