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.

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.

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 $d$.

Yes, there is one. As $f'$ represents the derivative of $f$, you can use the prime symbol like this:

$$(5x^3+7x^2+4x+9)' = 15x^2+14x+4$$

I have already seen it being used like that. Also note that as long as you don't make it confusing for the reader, you can make up your own notation if it's useful.

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