Verify the following equation using differentiation:

$\frac{d}{dx}(x^5-\ln (x^3+2))=5x^4-\frac{3x^2}{x^3+2}$

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. Plot the graph of the derivative of $R$ and determine the year when the population is growing fastest. Round the answer to 2 decimal places.

Find the value of $h^{\prime }\left(e\right)$ where $h\left(x\right)={\left(3x\right)}^{\frac{1}{3x}}$, and $e$ is the base of the natural logarithm.

Find the derivative of the function $f\left(x\right)={(3+\frac{1}{x})}^{2x}$ using logarithmic differentiation.

Find the derivative of $f\left(x\right)=\cos \left({\tan }^{-1}\left(10\ln x\right)\right)$.

A lighthouse stands $200$ meters tall on a straight coastline. A ship sails directly towards the lighthouse. Let $\theta$ be the angle of elevation of the lighthouse from the ship. Find the rate of change of the angle of elevation $\frac{d\theta }{dx}$ when the ship is $x=300\text{ m}$ away from the lighthouse.

A streetlight of height $H$ is fixed at point $A$ on the ground. A person walks away from the streetlight along a straight path at a constant speed. Let $B$ be the point where the person is currently standing, and let $C$ be the distance between points $A$ and $B$ on the ground. Define $\theta$ as the angle between the streetlight and the line connecting the top of the streetlight to point $B$. Find $\frac{d\theta }{dC}$.