I know that generally the impulse response of a system is the output when input is an impulse. It is also the differentiation of a step response. So if we have a system described by thi differential equation:
$$\frac{d^2 y(t)}{dt^2}+5\frac{\text{d}y(t)}{\text{d}t}
+6y(t)=-\frac{\text{d}x(t)}{\text{d}t}$$
where $x(t)$ is the input, $y(t)$ is the output, how do I find the impulse response?

My first idea was to let $x(t)$ be a ramp function so that it's derivative would be a step function then I could easily solve the differential equation and differentiate what I got to get the step response then differentiate it again to get the impulse response. But now I'm doubting this path as the ramp function is not a bounded function so the response might be unstable. I'm not sure how to approach this.