Say $y=f(x)$. To my understanding, the meaning of the derivative $\frac {dy} {dx} = f'(x)$ in English would be something like "the change in y with respect to the change in x at a given point is $f'(x)$."
After a symbolic manipulation of the derivative where we "multiply" both sides by $dx$, we get the differential $dy=f'(x)dx$. My question is, what does the differential actually mean (verbally), and how is it different from the derivative?
Just like $y$ is a function of $x$, $dy$ is also a function of $x$, and is the best linear approximation of $y$ at some point, $p$. However, whereas the tangent line passes through $(p,f(p))$, the function $dy(x)$ is shifted so that $(p,f(p)$ becomes the origin, so $dy(0)=0$. This is done so that $dy$ is a linear function in the linear algebra sense, i.e, $dy(c_1x_1+c_2x_2)=c_1dy(x_1)+c_2dy(x_2)$.
Since the tangent line is given by $f'(p)(x-p)+f(p)$, this means that $$dy(x)=f'(p)x,$$ or put in a more suggestive form, $dy(x-p)=f'(p)(x-p)$.
What about $dx$? Well, $x$ itself is a function of $x$ ($x=1\cdot x$), so $dx$ is the best linear approximation of that function at $p$, shifted so $(p,p)$ becomes the origin. But that function is itself linear, so $$dx(x)=x.$$
Note that means $\frac{dy}{dx}$ is also a function of $x$, being the quotient of two functions. What function is it? $$ \frac{dy(x)}{dx(x)}=\frac{f'(p)\cdot x}{x}=f'(p) $$ so $\frac{dy}{dx}$ is a constant, and that constant is $f'(p)$.
While the derivative at $x_0$ is the rate of change of $y$ with respect to $x$ , $\mathrm d\mkern1.5mu y$ is the best linear approximation of $y$ in a small neighbourhood of $x_0$.
Best is to be taken in a precise sense: it is the only linear approximation such that, in Landau notation: $f(x) -(f(x_0)+f'(x_0)\,\mathrm d\mkern 1.5mu x) $ is $\,o(\mathrm d\mkern 1.5mu x)$.
A differential is an infinitesimaly small change in a variable so $dy $ is the infinitesimal change in y at a given point... hence $dy=f'(x_0)dx $ is the change in y at $x_0$, infintesimaly speaking. This change can be approximated as $\delta y $ by doing a linear approximation $\delta y=f (x_0) \delta x $ or any higher order approximation according to Taylor series...quadratic, cubic.... The derivative is this change with respect to another change, again. Infinitesimaly speaking.
