What is the difference between a partial derivative and a total derivative? Why would I use one over the other? What kind of applications uses partial derivatives and what kind of applications uses total derivatives?
The correct answer and explanation is:
A partial derivative refers to the derivative of a function with respect to one variable while holding the other variables constant. It is commonly used when dealing with functions of multiple variables. For example, if you have a function f(x,y)f(x, y), the partial derivative with respect to xx is calculated while treating yy as a constant. Mathematically, it is written as ∂f∂x\frac{\partial f}{\partial x}.
In contrast, a total derivative takes into account the changes in all the variables that a function depends on. It considers how the function changes with respect to one variable while also accounting for the indirect influence of other variables that are themselves functions of that variable. For example, if f(x,y(t))f(x, y(t)) depends on xx and yy, which is a function of tt, the total derivative with respect to tt would be computed as ddtf(x,y(t))\frac{d}{dt} f(x, y(t)), applying the chain rule.
The choice between partial and total derivatives depends on the situation. Partial derivatives are used when a function explicitly depends on multiple independent variables, and you are interested in how the function changes with respect to one variable at a time while keeping the others fixed. This is common in fields like multivariable calculus, thermodynamics, and economics, where the analysis focuses on how a system responds to changes in one variable independently of others.
Total derivatives are typically used when variables are not independent of each other, and changes in one variable affect others. This is common in dynamics, fluid mechanics, and control theory, where the interactions between variables are important, and you are interested in how all changes collectively influence the system.