Determine all the first and second order partial derivatives of the function f(x, y) = log (ax + by).

To find the first and second order partial derivatives of the function \( f(x, y) = \log(ax + by) \), we will start by calculating the first order partial derivatives with respect to \( x \) and \( y \). Then, we will proceed to the second order partial derivatives.

First Order Partial Derivatives

The first order partial derivatives are found by differentiating the function with respect to one variable while treating the other variable as a constant.

Partial Derivative with Respect to x

To find \( \frac{\partial f}{\partial x} \), we apply the chain rule. The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = ax + by \).

• First, compute \( \frac{du}{dx} = a \).
• Now, apply the chain rule:

\[ \frac{\partial f}{\partial x} = \frac{1}{ax + by} \cdot a = \frac{a}{ax + by} \]

Partial Derivative with Respect to y

Next, we find \( \frac{\partial f}{\partial y} \) in a similar manner. Here, we treat \( x \) as a constant.

• Compute \( \frac{du}{dy} = b \).
• Then, apply the chain rule:

\[ \frac{\partial f}{\partial y} = \frac{1}{ax + by} \cdot b = \frac{b}{ax + by} \]

Second Order Partial Derivatives

Now that we have the first order partial derivatives, we can find the second order partial derivatives by differentiating the first order derivatives.

Second Order Partial Derivative with Respect to x

To find \( \frac{\partial^2 f}{\partial x^2} \), we differentiate \( \frac{\partial f}{\partial x} \) with respect to \( x \).

• Using the quotient rule, where \( u = a \) and \( v = ax + by \):

\[ \frac{\partial^2 f}{\partial x^2} = \frac{0 \cdot (ax + by) - a \cdot a}{(ax + by)^2} = \frac{-a^2}{(ax + by)^2} \]

Second Order Partial Derivative with Respect to y

Next, we compute \( \frac{\partial^2 f}{\partial y^2} \) by differentiating \( \frac{\partial f}{\partial y} \) with respect to \( y \).

• Again, applying the quotient rule:

\[ \frac{\partial^2 f}{\partial y^2} = \frac{0 \cdot (ax + by) - b \cdot b}{(ax + by)^2} = \frac{-b^2}{(ax + by)^2} \]

Mixed Partial Derivative

Finally, we find the mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \) by differentiating \( \frac{\partial f}{\partial x} \) with respect to \( y \).

• Using the quotient rule again:

\[ \frac{\partial^2 f}{\partial x \partial y} = \frac{0 \cdot (ax + by) - a \cdot b}{(ax + by)^2} = \frac{-ab}{(ax + by)^2} \]

Summary of Derivatives

To summarize, the derivatives of the function \( f(x, y) = \log(ax + by) \) are as follows:

• First order partial derivatives:
  • \( \frac{\partial f}{\partial x} = \frac{a}{ax + by} \)
  • \( \frac{\partial f}{\partial y} = \frac{b}{ax + by} \)
• Second order partial derivatives:
  • \( \frac{\partial^2 f}{\partial x^2} = \frac{-a^2}{(ax + by)^2} \)
  • \( \frac{\partial^2 f}{\partial y^2} = \frac{-b^2}{(ax + by)^2} \)
  • \( \frac{\partial^2 f}{\partial x \partial y} = \frac{-ab}{(ax + by)^2} \)

This comprehensive approach ensures that you understand how to derive both first and second order partial derivatives for the given function. If you have any further questions or need clarification on any steps, feel free to ask!