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

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 $d$.