Different between Partial and Ordinary derivative

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I took already Calculus and Ordinary differential equations but my fluids mechanics Professor ask us to write to pages about the difference between a partial and a ordinary derivative. The problem is that the only thing I know is that ordinary derivative are the differentiation of a function of one variable, and partial derivative the differentiation of a function of multiple variables about one variable.

This won't fill 2 pages. Somebody could help with some other differences?

 

[EnumaElish]

Comparing the total (ordinary) derivative of a function with multiple variables to the partial der. with resp. to one of the variables would be more illuminating and voluminous.

 

[tacman]

Sure, I'd be happy to help explain the differences between partial and ordinary derivatives.

First, let's review the definition of a derivative. A derivative is a mathematical tool used to measure the rate of change of a function with respect to its input variable. In other words, it tells us how much a function is changing at a specific point.

An ordinary derivative, also known as a total derivative, is the derivative of a function with respect to a single independent variable. It is denoted by the prime notation (') or by using the Leibniz notation (dy/dx). For example, if we have a function f(x) = x^2, the ordinary derivative would be f'(x) = 2x. This tells us how much the function is changing as x changes.

On the other hand, a partial derivative is the derivative of a function with respect to one of its many independent variables, while holding all other variables constant. It is denoted by the ∂ (partial) symbol. For example, if we have a function f(x,y) = x^2 + y^2, the partial derivative with respect to x would be ∂f/∂x = 2x, while the partial derivative with respect to y would be ∂f/∂y = 2y. This tells us how much the function is changing as x or y changes, while the other variable remains constant.

One key difference between partial and ordinary derivatives is the number of variables involved. Ordinary derivatives only have one independent variable, while partial derivatives have multiple. This means that partial derivatives can give us more information about how a function is changing, as it takes into account the effects of each independent variable.

Another difference is the interpretation of the derivative. With ordinary derivatives, we can interpret the result as the slope of the tangent line to the function at a specific point. However, with partial derivatives, we cannot do this as the function has multiple variables and cannot be represented on a two-dimensional graph.

Furthermore, partial derivatives are often used in multivariable calculus and in fields such as physics and engineering, where functions often have multiple independent variables. They are also used in optimization problems, where we need to find the maximum or minimum values of a function with multiple variables.

In summary, the main differences between partial and ordinary derivatives are the number of variables involved, the interpretation of the derivative, and their applications. I hope this helps you in your assignment