Finding the Fundamental Solution of the Operator $\frac{\partial}{\partial x}+c$

Question

I am trying to find the Fundamental Solution for the Operator $\frac{\partial}{\partial x}+c$. I approach it by considering a radial function $v(r)$, and considering the ODE $v'(r)+c v=0$. This ODE has the solution $v=\frac{A}{e^{cr}}$, where $A\in\mathbb R$. Then,take an interval centered at $x$ with width $2\epsilon$. Then: $$1="\int_{x-\epsilon}^{x+\epsilon}\delta(r)"=\int_{x-\epsilon}^{x+\epsilon}u'(r) dr+\int_{x-\epsilon}^{x+\epsilon}cu(r)dr$$ I am unsure as to how this leads to a Fundamental Solution. If anyone could guide me in the right direction it would be appreciated.