Re-scaling a delay integro-differential equation

Question

For the following delay differential equation: $$\frac{dy}{dt} = \int_{t-\delta}^t |y'(s)| ds,$$ I'd like to rescale $t$ to $\tau = t/\delta$. My attempt at this is as follows: Define $z(\tau)$ such that $y(t) = y(\tau \delta) = z(\tau)$. Then $$ \frac{dz}{d\tau} = \frac{dy}{d\tau} = \frac{dy}{dt} \frac{dt}{d\tau} = \delta \frac{dy}{dt} = \delta \int_{t-\delta}^t |y'(s)| ds = \delta \int_{\delta \tau-\delta}^{\delta \tau} |y'(s)| ds.$$ Now, since $y(s) = z(s/\delta)$ then $\frac{dy}{ds} = \frac{1}{\delta}\frac{dz}{ds}(s/\delta)$. So $$\frac{dz}{d\tau} = \delta \int_{\delta \tau-\delta}^{\delta \tau} |y'(s)| ds = \delta \int_{\delta (\tau - 1)}^{\delta \tau} \frac{1}{\delta} |z'(s/\delta)| ds$$. Finally, if we let $\sigma = s/\delta$ then $$\frac{dz}{d\tau} = \int_{\tau-1}^\tau |z'(\sigma)| d\sigma.$$ Is my working correct?