First timer here.
I would like to know the correct ways of writing and interpreting derivatives written using leibniz notation. I have a sample of 8 symbols below. For each one of them I would like to know:
- whether it is incorrect notation
- if it is not incorrect whether there is a better way to write the same
- what function is the derivative operator acting on
$$\begin{array}{rrrr}
\frac{\mathrm{d}x^2}{\mathrm{d}x}, &
\frac{\mathrm{d}kx^2}{\mathrm{d}x}, &
\frac{\mathrm{d}x^2+x^3}{\mathrm{d}x}, &
\frac{\mathrm{d}\left(x^2+x^3\right)}{\mathrm{d}x},\\
\frac{\mathrm{d}}{\mathrm{d}x}x^2, &
\frac{\mathrm{d}}{\mathrm{d}x}kx^2, &
\frac{\mathrm{d}}{\mathrm{d}x}x^2+x^3, &
\frac{\mathrm{d}}{\mathrm{d}x}\left(x^2+x^3\right)
\end{array}
$$
Best Answer
Of course notation is not completely unambiguous. But here is how I would interpret your examples.
$\frac{\mathrm{d}x^2}{\mathrm{d}x}$ should be differential of $x^2$ wrt $x$. Hence it should equal $2x$.
$\frac{\mathrm{d}kx^2}{\mathrm{d}x}$. I do not know what $k$ is supposed to be. I assume it is some constant, in this case I calculate this to equal $2kx$, the differential of $kx^2$ with respect to $x$. Although I suppose one could argue this equals $0$, interpreting the nominator as $(\mathrm d k)x^2$, and the differential of constants vanishes.
$\frac{\mathrm{d}x^2+x^3}{\mathrm{d}x}$. I'd say this doesn't make sense. To me $\mathrm d x$ should be a placeholder for something very small, and as $\mathrm d$ is not applied to $x^3$ (I'd think that addition should bind weaker than $\mathrm d$), I wouldn't know how to interpret $\frac {x^3}{\mathrm dx}$.
$\frac{\mathrm{d}\left(x^2+x^3\right)}{\mathrm{d}x}$ is just $2x + 3x^2$
$\frac{\mathrm{d}}{\mathrm{d}x}x^2$ This is the same as 1. to me.
$\frac{\mathrm{d}}{\mathrm{d}x}kx^2$ is the same as in 2.
$\frac{\mathrm{d}}{\mathrm{d}x}x^2+x^3$ is just $2x + x^3$.
$\frac{\mathrm{d}}{\mathrm{d}x} \left(x^2+x^3\right)$ should be $2x + 3x^2$.
The pattern is that $\mathrm d$ binds more or less like multiplication with a variable.