# Explanation of Total Differential vs Total Derivative

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.

• 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. – emonHR Dec 28 '19 at 12:36
• 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 $(dx,dy):=(\Delta x,\Delta y)=(h,k)$. – Matematleta Dec 28 '19 at 15:22

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

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