We consider the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ and the language $L=\{+, \cdot , \frac{d}{dx} , 0, 1\}$.

The ring consists of elements of the form $$\sum_{i=0}^N \alpha_i e^{\lambda_i x}$$ where $\alpha_i , \lambda_i \in \mathbb{C}$.

In the language there is no symbol $e^x$.

When we want to write a formula in the structure $$\left (\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}] ; +, \cdot , \frac{d}{dx} , 0, 1\right )$$ can we use the symbol $e^x$ because it is an element of the ring?

Or do we have to define it somehow using the operations of the language?

1more comment