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?
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.
Just as the symbol $$\int(\cdots)\;dx$$ denotes the antiderivative of something (the expression where the "$\cdots$" is), so the symbol $$\frac{d}{dx}(\cdots)$$ denotes the derivative of something (again, the expression where the "$\cdots$" is).

For example, you would have $$\frac{d}{dx}(13x^2-27x+1) = 26x-27$$
just as you would have $$\int(26x - 27)\;dx = 13x^2-27x + C$$

Occasionally "$D$" or "$D_x$" is seen in lieu of "$\frac{d}{dx}$", and it is very frequent to use an appended "prime" or "apostrophe" to mean the same thing, as $$(13x^2-27x+1)' = 26x-27$$

In other words, $$(\cdots)'\equiv \frac{d}{dx}(\cdots)$$