Q:

Is there a symbol for "taking a derivative of something"?

When presented with an equation, say, $y=5x^3+7x^2+4x+9$, you can write on the second line, $\frac{dy}{dx}=15x^2+14x+4$. Similarly, $f(x)=5x^3+7x^2+4x+9$ and $f'(x)=15x^2+14x+4$. But is there a way to write "the derivative of $5x^3+7x^2+4x+9$ is $15x^2+14x+4$" in just one line? 
What should l write, $\frac{dy}{d5x^3+7x^2+4x+9}=\cdots$? That fraction just gives me a headache trying to understand it. 
What about $f'(5x^3+7x^2+4x+9)=\cdots$? For all the reader knows, $f(x)$ could be anything, and the writer wanted them to plug in $5x^3+7x^2+4x+9$ into the original $f(x)$ and then take the derivative. 
So has anyone come up with a better way to write this that does not involve defining anything and then using the newly defined function/operator?

A:

You would denote the derivative of $5x^3+7x^2+4x+9$ as
$$\frac{d}{dx}(5x^3+7x^2+4x+9)$$
That is the only notation I've ever seen unless the expression is expressed as a function.

A:

A common choice of notation is $D_{x}(5x^3 + 7x^2 + 4x + 9)$. The subscript indicates the variable with respect to which one is differentiating. 

A:

The most common choice is $\frac{d}{dx}$.
If the variable is clear from context, you can use a plain $D$.
If you have several variables and you only want to differentiate with respect to one, it's best to write it as a partial derivative with $\frac{\partial}{\partial x}$ or $\partial_x$.
I have also seen notations like $(5x^3+7x^2+4x+9)'$ or $(5x^3+7x^2+4x+9)_x$, but I would strongly recommend using $\frac{d}{dx}$ instead.
There are several kinds of derivatives, and it's good to use notation that is compatible with them (uses similar syntax).
It is easy to replace $\frac{d}{dx}$ with a $\frac{\partial}{\partial x}$, a $\nabla$, a $\Delta$ or a $d$.

