91Ó°ÊÓ

Use implicit differentiation to find \(d y / d x\). $$x^{3}+y^{3}=18 x y$$

Find \(d y / d x\). $$y=\csc x-4 \sqrt{x}+\frac{7}{e^{x}}$$

A rock thrown vertically upward from the surface of the moon at a velocity of \(24 \mathrm{m} / \mathrm{sec}\) (about \(86 \mathrm{km} / \mathrm{h})\) reaches a height of \(s=24 t-0.8 t^{2} \mathrm{m}\) in \(t\) sec. a. Find the rock's velocity and acceleration at time \(t .\) (The acceleration in this case is the acceleration of gravity on the moon.) b. How long does it take the rock to reach its highest point? c. How high does the rock go? d. How long does it take the rock to reach half its maximum height? e. How long is the rock aloft?

Find the values. $$\cot \left(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$$

A cube's surface area increases at the rate of 72 in \(^{2} /\) sec. At what rate is the cube's volume changing when the edge length is \(x=3\) in?

A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of \(25 \mathrm{ft} / \mathrm{sec}\). How fast must she let out the string when the kite is 500 ft away from her?

Find the derivatives of the functions in Exercises \(23-50\). $$y=\frac{1}{x} \sin ^{-5} x-\frac{x}{3} \cos ^{3} x$$

What is the rate of change of the volume of a ball \(\left(V=(4 / 3) \pi r^{3}\right)\) with respect to the radius when the radius is \(r=2 ?\)

Find the derivatives of the function. $$y=\sqrt[3]{x^{46}}+2 e^{13}$$

Determine if the piecewise-defined function is differentiable at the origin. $$f(x)=\left\\{\begin{array}{ll}2 x-1, & x \geq 0 \\ x^{2}+2 x+7, & x<0\end{array}\right.$$