A pebble is dropped from a bridge, and the distance it falls is given by the equation $d\left(t\right)=15t^2$, where $d$ is in meters and $t$ is in seconds. If the pebble takes $7$ seconds to reach the water below, what is the height of the bridge? Also, calculate the pebble's speed upon hitting the water in kilometers per hour.

Determine the derivative of $g\left(x\right)=\frac{4x^4+3x^3-2x^2}{x^4}$.

Determine the first, second, and third derivatives of the function $g\left(x\right)=4x^3+7e^x$.

Find the second derivative $y^{\prime \prime }$ for the function $y=25\cot x$.

Find the slope of the curve $y=xh\left(x\right)$ at the point $(3,15)$ if $h\left(3\right)=5$ and $h^{\prime }\left(3\right)=6$.

Evaluate the following derivative using the graph given below.

$\frac{\text{d}}{\text{dx}}{\left(\frac{f\left(x\right)}{g\left(x\right)}\right)}_{x=2}$

Find the derivative of the function $f\left(x\right)=\frac{2\cos x+3\sin x}{2\cos x-3\sin x}$.

Consider the function $f\left(x\right)=\cos x$ on the interval $[0,2\pi ]$. Using the difference quotient $D\left(x\right)=\frac{f(x+0.02)-f\left(x\right)}{0.02}$, what does the graph of $D\left(x\right)$ resemble?

Determine the equation of the line tangent to the graph of $y=x\sqrt{17-x^2}$ at the point $(1,4)$.

A farmer is analyzing the growth pattern of a particular crop which can be approximated by the function $G\left(m\right)=5-2\cos \left(\frac{2\pi (m+15)}{365}\right)$, where $G$ represents the height of the crop in meters, and $m$ is the number of days since planting on March $1$ (considered as day $0$). What is the rate of growth of the crop with respect to days at any given day $m$?