A certain type of star emits energy according to the formula $E=150\cdot {10}^{0.5M}$, where $E$ is the energy in joules and $M$ is the magnitude of the star's brightness. Find $E^{\prime }\left(M\right)$ and approximate the value of $E^{\prime }\left(2\right)$ to a whole number. Provide an interpretation of this derivative.

What is the derivative of $q\left(x\right)=\frac{25}{3+5^{-x}}$?

Find the second derivative of the function $f\left(x\right)=\ln (3x^2+2)$.

In a newly established wildlife reserve, $100$ rabbits are introduced into an area with an estimated carrying capacity of $10,000$ rabbits. A logistic model of the rabbit population is given by $R(t) = \frac{1,000,000}{100 + 9900 e^{-0.3t}}$, where $t$ is measured in years. How fast (in rabbits per year) is the population growing at $t=0$? Approximate to the nearest whole number.