Understanding Total and Partial Derivatives in Multivariable Calculus

[Ted123]

Are the following equalities between total and partial derivatives true if \frac{dy}{dx}=f(x,y)? \displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y) \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)

 

[fauboca]

Are the following equalities between total and partial derivatives true if \frac{dy}{dx}=f(x,y)? \displaystyle \frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} f(x,y) \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)

This is a good resource
http://en.wikipedia.org/wiki/Total_derivative

 

[Ted123]

This is a good resource
http://en.wikipedia.org/wiki/Total_derivative

The first one is definitely true, but I'm not quite sure about the 2nd...