Partial Derivatives of ln(x+y)/(xy)

Thread starter littlesohi
Start date Oct 17, 2010
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Derivatives Partial Partial derivatives

In summary, a partial derivative is a measure of the rate of change of a function with respect to one variable while holding all other variables constant. To calculate a partial derivative, the function is differentiated with respect to the specific variable, treating all other variables as constants. This is done using various rules depending on the complexity of the function. In real life, partial derivatives are important in many fields such as economics, physics, engineering, and statistics, as they allow us to analyze how a function changes in response to changes in specific variables and are essential in optimization and gradient-based algorithms. They also have practical applications in determining maximum profit, analyzing relationships between physical quantities, and optimizing complex systems, as well as in data analysis and machine learning.

Oct 17, 2010

littlesohi

I need help with this one:

Find fxy in:

ln(x+y)/(xy) .. the ln applies to the whole problem.

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Oct 17, 2010

arildno

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littlesohi said:

I need help with this one:

Find fxy in:

ln(x+y)/(xy) .. the ln applies to the whole problem.

Well.

You might rewrite this using the prortyies of the logarithm, say:
f(x,y)=ln(x+y)-ln(xy)=ln(x+y)-ln(x)-ln(y)

1. What is the definition of a partial derivative?

A partial derivative is a mathematical concept used to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant.

2. How do you calculate a partial derivative?

To calculate a partial derivative, you take the derivative of the function with respect to the specific variable, treating all other variables as constants. This is done using the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function.

3. What is the partial derivative of ln(x+y)/xy?

The partial derivative of ln(x+y)/xy with respect to x is 1/(x(x+y)), and the partial derivative with respect to y is 1/(y(x+y)).

4. Why are partial derivatives important?

Partial derivatives are important in many areas of mathematics and science, including economics, physics, engineering, and statistics. They allow us to analyze how a function changes in response to changes in specific variables, and are essential in optimization and gradient-based algorithms.

5. How can partial derivatives be applied in real life?

Partial derivatives are used in various real-life applications, such as determining the maximum profit for a company based on different variables, analyzing the relationships between different physical quantities in physics, and optimizing the performance of complex systems in engineering. They can also be used in data analysis and machine learning algorithms.