91Ó°ÊÓ

Find the derivatives of all orders of the functions in Exercises \(41-44\) $$y=(x-1)(x+2)(x+3)$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\frac{t}{t+1}}$$

Have any horizontal tangent lines in the interval \(0 \leq x \leq 2 \pi ?\) If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher. $$y=\frac{\sec x}{3+\sec x}$$

Find the derivatives of the functions. $$f(\theta)=\left(\frac{\sin \theta}{1+\cos \theta}\right)^{2}$$

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

Find the derivatives of the functions. $$g(t)=\left(\frac{1+\sin 3 t}{3-2 t}\right)^{-1}$$

Determine whether the piecewise-defined function is differentiable at \(x=0\). $$g(x)=\left\\{\begin{array}{ll}2 x-x^{3}-1, & x \geq 0 \\ x-\frac{1}{x+1}, & x<0\end{array}\right.$$

Ships Two ships are steaming straight away from a point \(O\) along routes that make a \(120^{\circ}\) angle. Ship \(A\) moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yd). Ship \(B\) moves at 21 knots. How fast are the ships moving apart when \(O A=5\) and \(O B=3\) nautical miles?

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\frac{1}{t(t+1)}}$$

Find the normal lines to the curve \(x y+2 x-y=0\) that are parallel to the line \(2 x+y=0\).