Partial Derivative Analysis Question

In summary, a partial derivative is a mathematical concept used in multivariable calculus to measure the rate of change of a function with respect to one of its variables while holding the other variables constant. This analysis is important in fields such as physics, economics, and engineering, as it allows us to understand how a function responds to changes in its variables. A partial derivative is calculated by taking the derivative of a function with respect to one variable while treating the others as constants. It differs from a total derivative, which measures the overall rate of change of a function with respect to all variables. Partial derivatives have various real-life applications, including determining maximum and minimum values in economics, optimizing processes in engineering, and predicting the behavior of a system in physics.
  • #1
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Homework Statement

Given a graph of f(x,y), how can you determine where the partial and second derivatives are positive, negative, or zero?

The attempt at a solution

The first partial derivative is fairly easy to picture so I'm more concerned about the second partial derivative. I'm having troubles imagining what this would look like and an explanation could come of use. Thanks in advance for the help.
 
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  • #2
Take a loot at the notion of an "inflexion point" for a function of one variable. This should give you some intuition.
 

1. What is a partial derivative?

A partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables while holding the other variables constant. It is denoted by ∂ (pronounced "partial") and is often used in multivariable calculus to analyze how a function changes in relation to multiple variables.

2. Why is partial derivative analysis important?

Partial derivative analysis allows us to understand how a function changes in response to changes in its variables. This is especially useful in fields such as physics, economics, and engineering, where many variables can affect the outcome of a system. It also helps us find critical points and determine the behavior of a function in different directions.

3. How is a partial derivative calculated?

A partial derivative is calculated by taking the derivative of a function with respect to one of its variables, while treating the other variables as constants. This means that all terms in the function that do not contain the variable of interest will be treated as constants and their derivative will be zero. The resulting derivative is the partial derivative with respect to that variable.

4. What is the difference between a partial derivative and a total derivative?

A partial derivative measures the rate of change of a function with respect to one variable, while holding all other variables constant. On the other hand, a total derivative measures the overall rate of change of a function with respect to all of its variables. In other words, a partial derivative is a special case of a total derivative, where all other variables are held constant.

5. How are partial derivatives used in real-life applications?

Partial derivatives are used in various real-life applications, such as determining the maximum and minimum values of a function in economics, optimizing processes in engineering, and predicting the behavior of a system in physics. They are also used in machine learning and data analysis to understand how changing one variable affects the outcome of a model.

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