Chapter 3

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

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\tan ^{-1} \sqrt{x^{2}-1}+\csc ^{-1} x, \quad x>1$$

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=x^{2}+2 x, \quad x_{0}=1, \quad d x=0.1$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \left(\frac{\left(x^{2}+1\right)^{5}}{\sqrt{1-x}}\right)$$

Find the derivatives of the functions in Exercises \(23-50\). $$h(x)=x \tan (2 \sqrt{x})+7$$

Do the graphs of the functions have any horizontal tangents 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. $$y=x+\sin x$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cot ^{-1} \frac{1}{x}-\tan ^{-1} x$$

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=2 x^{2}+4 x-3, x_{0}=-1, \quad d x=0.1$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \sqrt{\frac{(x+1)^{5}}{(x+2)^{20}}}$$

Find the derivatives of the functions in Exercises \(23-50\). $$k(x)=x^{2} \sec \left(\frac{1}{x}\right)$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$x^{2} \cos ^{2} y-\sin y=0, \quad(0, \pi)$$

Do the graphs of the functions have any horizontal tangents 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. $$y=2 x+\sin x$$

Find the derivatives of the function. $$r=e^{\theta}\left(\frac{1}{\theta^{2}}+\theta^{-\pi / 2}\right)$$

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

A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in \(^{3} / \mathrm{min}\), how fast is the thickness of the ice decreasing when it is 2 in. thick? How fast is the outer surface area of ice decreasing?

Find the derivative of \(y\) with respect to the appropriate variable. $$y=x \sin ^{-1} x+\sqrt{1-x^{2}}$$

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=x^{3}-x, \quad x_{0}=1, \quad d x=0.1$$

Find the derivatives of the functions in Exercises \(23-50\). $$f(x)=\sqrt{7+x \sec x}$$

Find the two points where the curve \(x^{2}+x y+y^{2}=7\) crosses the \(x\) -axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents?

Do the graphs of the functions have any horizontal tangents 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. $$y=x-\cot x$$

Find the derivatives of all orders of the functions. $$y=\frac{x^{2}}{2}-\frac{3}{2} x^{2}-x$$

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

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\left(x^{2}+1\right)(x-1)^{2}}$$

A highway patrol plane flies 3 mi above a level, straight road at a steady \(120 \mathrm{mi} / \mathrm{h}\). The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi, the line-of-sight distance is decreasing at the rate of \(160 \mathrm{mi} / \mathrm{h}\). Find the car's speed along the highway.

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \left(x^{2}+4\right)-x \tan ^{-1}\left(\frac{x}{2}\right)$$

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=x^{4}, \quad x_{0}=1, \quad d x=0.1$$

Find the derivatives of the functions in Exercises \(23-50\). $$g(x)=\frac{\tan 3 x}{(x+7)^{4}}$$

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

Do the graphs of the functions have any horizontal tangents 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. $$y=x+2 \cos x$$

Find the derivatives of all orders of the functions. $$y=\frac{x^{5}}{120}$$

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

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

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=x^{-1}, \quad x_{0}=0.5, \quad d x=0.1$$

Find all points on the curve \(y=\tan x,-\pi / 2

Find the derivatives of all orders of the functions. $$y=(x-1)(x+2)(x+3)$$

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

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?

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=x^{3}-2 x+3, x_{0}=2, \quad d x=0.1$$

Find the derivatives of the functions in Exercises \(23-50\). $$g(t)=\left(\frac{1+\sin 3 t}{3-2 t}\right)^{-1}$$

Find all points on the curve \(y=\cot x, 0

Find the derivatives of all orders of the functions. $$y=\left(4 x^{2}+3\right)(2-x) x$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sin \theta) \sqrt{\theta+3}$$

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\tan ^{-1} 2\) b. \(\cos ^{-1} 2\)

At what rate is the angle between a clock's minute and hour hands changing at 4 o'clock in the afternoon?

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=(4 / 3) \pi r^{3}\) of a sphere when the radius changes from \(r_{0}\) to \(r_{0}+d r\)

Find the derivatives of the functions in Exercises \(23-50\). $$r=\sin \left(\theta^{2}\right) \cos (2 \theta)$$

Find the first and second derivatives of the functions. $$y=\frac{x^{3}+7}{x}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\tan \theta) \sqrt{2 \theta+1}$$

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\csc ^{-1}(1 / 2)\) b. \(\csc ^{-1} 2\)

An explosion at an oil rig located in gulf waters causes an elliptical oil slick to spread on the surface from the rig. The slick is a constant 9 in. thick. After several days, when the major axis of the slick is 2 mi long and the minor axis is \(3 / 4\) mi wide, it is determined that its length is increasing at the rate of \(30 \mathrm{ft} / \mathrm{hr},\) and its width is increasing at the rate of \(10 \mathrm{ft}\) hr. At what rate (in cubic feet per hour) is oil flowing from the site of the rig at that time?