Given z = ln ( y x ) , find ? f ? x and ? f ? y .


Given {eq}z = \ln (\frac{y}{x}) {/eq}, find {eq}\frac{\partial f}{\partial x} {/eq} and {eq}\frac{\partial f}{\partial y} {/eq}.

Answer and Explanation:

Applying the properties of the logarithms, we have:

{eq}z = \ln \left( {\frac{y}{x}} \right) = \ln y - \ln x {/eq}

The partial derivatives can be easily written:

{eq}\frac{{\partial z}}{{\partial x}} = - \frac{1}{x}\\ \frac{{\partial z}}{{\partial y}} = \frac{1}{y} {/eq}

Learn more about this topic:

Applying the Rules of Differentiation to Calculate Derivatives

from Math 104: Calculus

Chapter 9 / Lesson 13

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