## Calculus (3rd Edition)

$$Q_L= \left(\frac{1}{M}-\frac{tL}{M^2}\right)e^{-Lt/M},\\ Q_M =-\left(\frac{L}{M^2}-\frac{L^2t}{M^3}\right)e^{-Lt/M},\\ Q_t=-\frac{L^2}{M^2}e^{-Lt/M}.$$
Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(e^x)'=e^x$ Since $Q=\frac{L}{M}e^{-Lt/M}$, then by the product rule, we have $$Q_L= \frac{1}{M}e^{-Lt/M}+ \frac{L}{M}e^{-Lt/M}\frac{-t}{M}=\left(\frac{1}{M}-\frac{tL}{M^2}\right)e^{-Lt/M},\\ Q_M= - \frac{L}{M^2}e^{-Lt/M}+ \frac{L}{M}e^{-Lt/M}\frac{Lt}{M^2}=-\left(\frac{L}{M^2}-\frac{L^2t}{M^3}\right)e^{-Lt/M},\\ Q_t=-\frac{L^2}{M^2}e^{-Lt/M}.$$