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So, I understand the total derivative is used when you cannot hold a variable constant, for example when a variable is defined by other variables that do not feature in the original equation. For example if you had:

$f(x,y) = 2x + 3y , x = x(r,w) , y = y(r,w)$, you could calculate the total derivatives of the function $f$ with respect to the independant variables $r,w$ rather than the partial derivatives of the dependent variables $x,y$.

Now, I want to know what the difference between that is and the total differentials, intuitively. I see that I can calculate the total differential of a function $f = f(x,y,z) = xy + 2yz + z -4z^{1/2} = 7$ as $f_x dx + f_y dy + f_z dz = 0$ whereby $f_x$ is the partial derivative of $f$ w.r.t $x$. But what is this 'total differential' actually telling me? Could someone verify my understandings and explain why/ when this total differential is used and why/when a total derivative is used? Many thanks.

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  • $\begingroup$ The total derivative of a function at a point is the $\color{red}{\text{best linear approximation}}$ near this point of the function with respect to its arguments. $\endgroup$ – emonHR Dec 28 '19 at 12:36
  • $\begingroup$ Do you know the definition of derivative for real-valued functions of several variables? If you do, then the total differential pops out as the action of the derivative on the vector $(dx,dy):=(\Delta x,\Delta y)=(h,k)$. $\endgroup$ – Matematleta Dec 28 '19 at 15:22
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Do you know the definition of derivative for real-valued functions of two variables? Fix $(x_0,y_0)\in \mathbb R^2$. Then, $f$ is $\textit{differentiable at}\ (x_0,y_0)$ if there exist a linear transformation $L:\mathbb R^2\to \mathbb R$ such that

$\tag 1 f(x_0+h,y_0+k)-f(x_0,y_0)=L(h,k)+r(h,k)\ \text{such that}\ \underset {(h,k)\to (0,0)}\lim \frac{r(h,k)}{h}=0$.

When you apply this definition, you get

$\tag 2 L(h,k)=\left(\frac{\partial f}{\partial x}\right)_{(x_0,y_0)}h+\left(\frac{\partial f}{\partial y}\right)_{(x_0,y_0)}k$

Now, changing notation, we see that the total differential pops out as the action of the derivative on the vector $(dx,dy):=(\Delta x,\Delta y)=(h,k)$, and so the image of the derivative is the equation of the tangent plane to $f$ at the point $(x_0,y_0)$, which provides an approximation to $f$ itself in a presumably small neighborhood of $(x_0,y_0))$. More precisely, from $(1)$ and $(2)$, we get

$\tag3 f(x_0+h,y_0+k)-f(x_0,y_0)=\left(\left(\frac{\partial f}{\partial x}\right)_{(x_0,y_0)}h+\left(\frac{\partial f}{\partial y}\right)_{(x_0,y_0)}k\right)+r(h,k).$

The condition on $r$ proves the claim, and now setting the term in parentheses to

$df$, and $f(x_0+h,y_0+k)-f(x_0,y_0)$ to $\Delta f,$

we get the more familiar expression

$\tag4 \Delta f\approx df\ \text{when}\ \Delta x\ \text{and}\ \Delta y\ \text{are small}$

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