Partial Differential

Question

If f is a function of x. Some times people write:

Is this a rule? does it have a proof? need help.

Thanks.

Answer 1

\begin{align*} x_i &= x_i(t) \text{ and}\\ F &= f(x_1, x_2, ..., x_n)\\ \frac{dF}{dt} &= \sum^n_{i=1} \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}\\ dF &= \sum^n_{i=1} \frac{\partial f}{\partial x_i} dx_i \end{align*}

In your case, F is a function of x, let x be the $x_1$, that is, $F = f(x_1)$, so \begin{align*} dF &=\frac{\partial f}{\partial x_1} dx_1 \\ &= \frac{df}{dx_1} dx_1 \end{align*}

Answer 2

$df=\frac{\partial f}{\partial x}dx$ is false if $f$ is a function of several variables (which is suggested by your question about partial derivatives). For example in case of $f(x,y)$ : $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$ If $f$ is function of one variable only, the "partial derivative" is no longer partial but is the usual derivative $$df=f'(x)dx=\left(\frac{\text{d}f}{\text{d}x}\right)dx$$ $\left(\frac{\text{d}f}{\text{d}x}\right)$ denotes a function, not a fraction.

In this case of one variable only, sometimes $\frac{\partial f}{\partial x}$ is loosely confused with $\frac{\text{d}f}{\text{d}x}$. That is a matter of conventional symbols.