The discussion focuses on the relationship between total and partial derivatives in multivariable calculus, specifically examining the equalities involving the function \( f(x,y) \) and its derivatives. The first equality, \( \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y) \), is confirmed as true. However, there is uncertainty regarding the second equality, \( \frac{d^2f}{dx^2} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} + \left( \frac{\partial f}{\partial y} \right) ^2 f(x,y) \), indicating a need for further clarification. Resources from Wikipedia on total derivatives are recommended for additional context.
PREREQUISITESStudents and professionals in mathematics, engineering, and physics who are looking to deepen their understanding of multivariable calculus and the application of derivatives in complex functions.
Ted123 said:Are the following equalities between total and partial derivatives true if [itex]\frac{dy}{dx}=f(x,y)[/itex]? [tex]\displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y)[/tex] [tex]\displaystyle \frac{d^2f}{dx^2} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} + \left( \frac{\partial f}{\partial y} \right) ^2 f(x,y)[/tex]
fauboca said:This is a good resource
http://en.wikipedia.org/wiki/Total_derivative