91Ó°ÊÓ

If \(x^{3}+y^{3}=16,\) find the value of \(d^{2} y / d x^{2}\) at the point (2,2).

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\sec ^{-1} \frac{1}{t}, \quad 0 < t < 1$$

Find \(d y\) $$y=3 \csc (1-2 \sqrt{x})$$

A draining hemispherical reservoir Water is flowing at the rate of \(6 \mathrm{m}^{3} / \mathrm{min}\) from a reservoir shaped like a hemispherical bowl of radius \(13 \mathrm{m},\) shown here in profile. Answer the following questions, given that the volume of water in a hemispherical bowl of radius \(R\) is \(V=(\pi / 3) y^{2}(3 R-y)\) when the water is \(y\) meters deep. a. At what rate is the water level changing when the water is 8 m deep? b. What is the radius \(r\) of the water's surface when the water is \(y\) m deep? c. At what rate is the radius \(r\) changing when the water is \(8 \mathrm{m}\) deep? CANT COPY THE GRAPH

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\ln x)$$

Object dropped from a tower An object is dropped from the top of a \(100-\mathrm{m}\) -high tower. Its height above ground after \(t\) sec is \(100-4.9 t^{2} \mathrm{m} .\) How fast is it falling 2 sec after it is dropped?

Find \(d p / d q\). $$p=\frac{\sin q+\cos q}{\cos q}$$

Find the derivatives of the functions in Exercises \(17-40 .\) $$y=2 e^{-x}+e^{3 x}$$

Find the derivatives of the functions. $$y=x^{2} \sin ^{4} x+x \cos ^{-2} x$$

Speed of a rocket At \(t\) sec after liftoff, the height of a rocket is \(3 t^{2} \mathrm{ft} .\) How fast is the rocket climbing 10 sec after liftoff?