The difference rule of derivatives is actually derived in differential calculus from first principle. For example, $f{(x)}$ and $g{(x)}$ are two differentiable functions and the difference of them is written as $f{(x)}-g{(x)}$. The derivative of difference of two functions with respect to $x$ is written in the following mathematical form.
$\dfrac{d}{dx}{\, \Big(f{(x)}-g{(x)}\Big)}$
Take $m{(x)} = f{(x)}-g{(x)}$ and then $m{(x+\Delta x)} = f{(x+\Delta x)}-g{(x+\Delta x)}$
According to definition of the derivative, write the derivative of the function $m{(x)}$ with respect to $x$ in limiting operation.
$\dfrac{d}{dx}{\, \Big(m{(x)}\Big)}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{m{(x+\Delta x)}-m{(x)}}{\Delta x}}$
Replace the actual functions of $m{(x)}$ and $m{(x+\Delta x)}$.
$\implies$ $\dfrac{d}{dx}{\, \Big(f{(x)}-g{(x)}\Big)}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{\Big(f{(x+\Delta x)}-g{(x+\Delta x)}\Big)-\Big(f{(x)}-g{(x)}\Big)}{\Delta x}}$
Now, take $\Delta x = h$ and start simplifying this function for deriving the derivative of difference of two functions by first principle.
$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{\Big(f{(x+h)}-g{(x+h)}\Big)-\Big(f{(x)}-g{(x)}\Big)}{h}}$
$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-g{(x+h)}-f{(x)}+g{(x)}}{h}}$
$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-f{(x)}-\Big(g{(x+h)}-g{(x)}\Big)}{h}}$
$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \Bigg[\dfrac{f{(x+h)}-f{(x)}}{h}-\dfrac{g{(x+h)}-g{(x)}}{h}\Bigg]}$
As per difference rule of limits, the limit of difference of two functions can be written as difference of their limits.
$=\,\,\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f{(x+h)}-f{(x)}}{h}}$ $-$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{g{(x+h)}-g{(x)}}{h}}$
According to first principle of differentiation, each term in the right right-hand side of the equation represents the derivative of the respective function.
$\,\,\, \therefore \,\,\,\,\,\,$ $\dfrac{d}{dx}{\, \Big(f{(x)}-g{(x)}\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $-$ $\dfrac{d}{dx}{\, g{(x)}}$
In this way, the difference rule of derivatives can be derived in differential calculus mathematically from first principle.
The derivative difference rule is also written in two forms alternatively by taking $u = f{(x)}$ and $v = g{(x)}$.
$(1) \,\,\,$ $\dfrac{d}{dx}{\, (u-v)}$ $\,=\,$ $\dfrac{du}{dx}$ $-$ $\dfrac{dv}{dx}$
$(2) \,\,\,$ ${d}{\, (u-v)}$ $\,=\,$ $du-dv$
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