$\displaystyle \frac{dM}{dt} = \frac{dM}{du} \cdot \frac{du}{dt}$.

Your expression for $\displaystyle \frac{dM}{du}$ is correct and it's value at $\displaystyle t = 0 \Rightarrow u = 1$ is correct.

However, you clearly need to multiply that result by the value of $\displaystyle \frac{du}{dt}$ at $\displaystyle t = 0$. Since this multiplying factor is equal to 1, you did in fact get the correct final answer - but by luck ....