Multi-variable Calculus : Partial differentiation

Thread starter smart_worker
Start date Nov 4, 2014
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Calculus Differentiation Multi-variable Partial Partial differentiation

In summary, partial differentiation is a mathematical concept used to find the rate of change of a function with respect to one of its variables while holding all other variables constant. It differs from ordinary differentiation in that it involves finding the rate of change with respect to one variable while keeping all other variables constant. The purpose of partial differentiation is to help us understand functions with multiple variables and find maximum and minimum values, making it useful in optimization problems. It has various real-world applications, such as in economics, physics, and engineering. To find partial derivatives, we treat all other variables as constants and differentiate the function with respect to the variable of interest, repeating the process for each variable to obtain a set of partial derivatives. These derivatives can then be used to

Nov 4, 2014

smart_worker

Homework Statement

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2. The attempt at a solution
By chain rule,

which simpifies to,

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After this I am struck.

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Nov 4, 2014

ShayanJ

Insights Author

Gold Member

Take into account [itex]\frac{\partial}{\partial u}=\frac{\partial x}{\partial u}\frac{\partial}{\partial x} + \frac{\partial y}{\partial u}\frac{\partial}{\partial y} [/itex] and similarly for v.

Related to Multi-variable Calculus : Partial differentiation

1. What is partial differentiation?

Partial differentiation is a mathematical concept used to find the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is commonly used in multi-variable calculus to analyze the behavior of functions with more than one independent variable.

2. How is partial differentiation different from ordinary differentiation?

Ordinary differentiation involves finding the rate of change of a function with respect to a single independent variable. Partial differentiation, on the other hand, involves finding the rate of change with respect to one variable while keeping all other variables constant.

3. What is the purpose of partial differentiation?

The purpose of partial differentiation is to help us understand how a function behaves when more than one variable is involved. It also allows us to find the maximum and minimum values of a function with multiple variables, which is useful in optimization problems.

4. Can you give an example of a real-world application of partial differentiation?

Partial differentiation has many real-world applications, such as in economics, physics, and engineering. For example, in economics, partial differentiation can be used to analyze the relationship between two or more variables, such as demand and price, in order to make informed decisions about production and pricing strategies.

5. What is the process for finding partial derivatives?

To find a partial derivative, we treat all other variables as constants and differentiate the function with respect to the variable of interest. This process is repeated for each variable in the function, resulting in a set of partial derivatives. These derivatives can then be used to analyze the behavior of the function and solve for critical points.