What is the difference between standard derivative and partial derivative?

Question

This is actually a very old question that now I have to face it again and look for answer of it. Suppose $f:\mathbb{R}^n\to \mathbb{R}, f(x_1,x_2,...,x_n)=y$ is a function. What is the difference between:

• $\frac{\partial f}{\partial x_i}$
• $\frac{df}{dx_i}$ if this means anything at all?

I am reading this book and the following passage is part of the book:

...it would be appropriate to introduce a scaled time $\tau$ via

$$\tau=\epsilon^2 t$$

and regard $u$ as depending both on $t$ and $\tau$, and having no explicit dependence on $\epsilon$; $t$ and $\tau$ will be treated as mutually independent. Correspondingly, the time differentiation should be transformed as

$$\frac{d}{dt}\to\frac{\partial}{\partial t}+\epsilon^2 \frac{\partial}{\partial \tau} $$

Where $u=X-X_0$ and $X_0$ is a stable answer for following differential equation: $$\frac{dX}{dt}=F(X)$$

Answer 1

$$\frac{\partial f}{\partial x_i}$$

Is a partial derivate so you suppose that all variables except $x_i$ are constant (you can this $f$ as a single variable function).

$$\frac{df}{dx_i}$$

Is a total derivate (all variables may vary). So you have to apply the chain rule and you get:

$$\frac{df}{dx_i}=\sum_j \frac{\partial f}{\partial x_j}\frac{dx_j}{dx_i}$$