quasar987 said:Mh, why is that?
I thought that by definition, dF is the formal expression
[tex]dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy[/tex]
So existence of partial derivatives is sufficient to make sense of dF.
A differential is an infinitesimal change in the value of a function, while a derivative is the instantaneous rate of change of a function at a specific point.
The differential of a multivariable function is calculated by taking the sum of the partial derivatives of the function with respect to each variable, multiplied by the corresponding infinitesimal change in each variable.
To find the derivative of a multivariable function, you must take the partial derivative of the function with respect to each variable, and then combine them using the chain rule or product rule if necessary.
Differentials and derivatives are closely related in multivariable functions, as the differential is essentially the linear approximation of the function at a specific point, and the derivative is the slope of the function at that point.
No, differentials and derivatives have different interpretations and uses in multivariable functions. Differentials are used for approximations and error analysis, while derivatives are used to find rates of change and optimize functions.