I am concerned with trying to understand the relationship between the total differential and the total derivative.
The definition for total differential of a function $y=f(x_{1},x_{2})$ that I have been given is $dy=f_{1}dx_{1}+f_{2}dx_{2}$ where $f_{i}$ denotes the partial derivative of $f$ wrt $x_{i}$ and $dx_{1}=\Delta x_{1}$ and $dx_{2}=\Delta x_{2}$. For small $dx_{i}$, $dy \approx \Delta y=f(x_{1}+dx_{1},x_{2}+dx_{2})-f(x_{1},x_{2})$.
The total derivative of $y$ wrt $x_{1}$ is given by $\frac{dy}{dx_{1}}=f_{1}+f_{2}\frac{dx_{2}}{dx_{1}}$, which is as if we took the total differential and divided it by $dx_{1}$.
Now, I know that the derivative of a function of one variable $w$ wrt $x$, denoted $\frac{dw}{dx}$, would be given by $\lim_{\Delta x \rightarrow 0}\frac{\Delta y}{\Delta x}$.
Would it be correct for me to say that we can arrive at the total derivative from the total differential, but that dividing the expression of the total differential by $dx_{1}=\Delta x_{1}$ does not give the total derivative, rather, you get the total derivative if you divide the total differential by $\Delta x_{1}$ and take the limit as $\Delta x_{1} \rightarrow 0$?
Thank you.
The differential of a function is the linear part of its variation wrt. the independent variables. This means that the expression
$$df=f_xdx+f_ydy$$ is exact for any finite value of $dx$ and $dy$, and so is
$$\frac{df}{dx}=f_x+f_y\frac{dy}{dx}.$$
On another hand, by the chain rule,
$$\lim_{\Delta x\to0}\frac{f(x+\Delta x,y(x+\Delta x))-f(x,y(x))}{\Delta x}=f_x+f_y\,y'$$ where the prime denotes the derivative on $x$,
$$y'=\frac{dy}{dx}.$$