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$\ln(x) + \ln(y) - \ln(xy)$

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Find derivatives of logarithmic functions

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Q.

\ln(x) + \ln(y) - \ln(xy)

Apply Logarithm Properties: Apply the properties of logarithms to simplify the function. The properties of logarithms tell us that $\ln(a) + \ln(b) = \ln(ab)$ and $\ln(a/b) = \ln(a) - \ln(b)$ . Using these properties, we can simplify the function $f(x, y)$ as follows: $f(x, y) = \ln(x) + \ln(y) - \ln(xy) = \ln(x) + \ln(y) - \ln(x) - \ln(y) = 0$ .
Derivative of Constant Function: Since the simplified function $f(x, y)$ is a constant ( $0$ ), its derivative with respect to $x$ or $y$ is also a constant. The derivative of a constant is $0$ . Therefore, the partial derivatives of $f(x, y)$ with respect to $x$ and $y$ are: $\partial f/\partial x = 0$ and $\partial f/\partial y = 0$ .
Write Gradient of Function: Combine the partial derivatives to write the gradient of $f(x, y)$ . The gradient of $f(x, y)$ is the vector of its partial derivatives, which in this case is: \(\newlineabla f(x, y) = (0, 0)\).

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