What do the symbols d/dx and dy/dx mean?

Solution 1:

The symbol $$ \frac{dy}{dx} $$ means the derivative of $y$ with respect to $x$. If $y = f(x)$ is a function of $x$, then the symbol is defined as $$ \frac{dy}{dx} = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}. $$ and this is is (again) called the derivative of $y$ or the derivative of $f$. Note that it again is a function of $x$ in this case. Note that we do not here define this as $dy$ divided by $dx$. On their own $dy$ and $dx$ don't have any meaning (here). We take $\frac{dy}{dx}$ as a symbol on its own that can't be slit up into parts.

The symbol $$ \frac{d}{dx} $$ you can consider as an operator. You can apply this operator to a (differentiable) function. And you get a new function. So if $f$ is a (differentiable) function that it makes sense to "apply" $\frac{d}{dx}$ to $f$ and write $$ \frac{d}{dx}f $$ If you write $y = f(x)$, then this is the same as $$ \frac{d}{dx}y = \frac{dy}{dx}. $$

Solution 2:

$\frac{d}{dx}$ means differentiate with respect to $x$.

$\frac{dy}{dx}$ means differentiate $y$ with respect to $x$.

Do you have any concrete examples for which you need to calculate these two? It would probably make it more easy to grasp for you if I could explain it in a few examples.

Solution 3:

$d f$ means the differential of function $f$. By definition $(df)(x) = \lambda t\in\mathbb{R}:f'(x)\cdot t$. In other words, differential is the linear function (of an additional variable denoted $t$ here) whose tangent is the derivative of $f$.

$d$ alone means the differential operator (a function of argument $f$).

Exercise: Show that $\frac{df}{dx}=f'$.