Partial Derivatives: Proving & Evaluating at (0,0)

In summary, partial derivatives are a type of derivative that measures the rate of change of a multivariable function with respect to one of its input variables while holding the other variables constant. To prove a partial derivative at a specific point, the definition of a partial derivative is used, involving taking a limit. A regular derivative measures the rate of change with respect to a single input variable, while a partial derivative measures the rate of change with respect to one input variable while holding the other variables constant. Partial derivatives are used in various real-world applications, such as optimization problems, machine learning algorithms, and calculations in physics, economics, and engineering. They can be evaluated at any point as long as the function is differentiable, but the value may vary
  • #1
SANGHERA.JAS
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0
Do I need to use Schwarz's or Young's theorems to prove it, if don't then do I need to evaluate them on (0,0) using definition.
 

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  • #2
I would start by taking the mixed partials (fxy and fxy) for each of the given functions, and evaluating these partials at the origin.
 

1. What are partial derivatives?

Partial derivatives are a type of derivative in multivariable calculus that measures the rate of change of a function with respect to one of its input variables while holding the other variables constant.

2. How do you prove a partial derivative at a specific point?

To prove a partial derivative at a specific point, you can use the definition of a partial derivative, which involves taking a limit of the function as the input variable approaches the desired point. If the limit exists, then the partial derivative exists at that point.

3. What is the difference between a partial derivative and a regular derivative?

A regular derivative measures the rate of change of a function with respect to a single input variable, whereas a partial derivative measures the rate of change with respect to one input variable while holding the other variables constant. This is necessary in multivariable functions, as there can be multiple input variables that affect the output.

4. Can you evaluate a partial derivative at any point?

Yes, you can evaluate a partial derivative at any point as long as the function is differentiable at that point. However, the value of the partial derivative may change depending on the point at which it is evaluated.

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

Partial derivatives are used in many fields of science and engineering, such as physics, economics, and engineering. They are particularly useful in optimization problems, where the goal is to find the maximum or minimum of a multivariable function. They are also used in gradient descent algorithms for machine learning and in calculating rates of change in thermodynamics and fluid dynamics.

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