?Towing a boat A boat is towed toward a dock by a cable attached to a winch that stands

State the derivative formulas for \(\sin ^{-1} x, \tan ^{-1} x\), and \(\sec ^{-1} x\).

What is the slope of the line tangent to the graph of \(y=\sin ^{-1} x\) at x = 0?

What is the slope of the line tangent to the graph of \(y=\tan ^{-1} x\) at x = -2?

How are the derivatives of \(\sin ^{-1} x\) and \(\cos ^{-1} x\) related?

If f is a one-to-one function with f(2) = 8 and \(f^{\prime}(2)=4\), what is the value of \(\left(f^{-1}\right)^{\prime}(8)\)?

Explain how to find \(\left(f^{-1}\right)^{\prime}\left(y_{0}\right)\), given that \(y_{0}=f\left(x_{0}\right)\).

Derivatives of inverse sine Evaluate the derivatives of the following functions. \(f(x)=\sin ^{-1}(2 x)\)

Derivatives of inverse sine Evaluate the derivatives of the following functions. \(f(x)=x \sin ^{-1} x\)

Derivatives of inverse sine Evaluate the derivatives of the following functions. \(f(w)=\cos \left[\sin ^{-1}(2 w)\right]\)

Derivatives of inverse sine Evaluate the derivatives of the following functions. \(f(x)=\sin ^{-1}(\ln x)\)

Derivatives of inverse sine Evaluate the derivatives of the following functions. \(f(x)=\sin ^{-1}\left(e^{-2 x}\right)\)

Derivatives of inverse sine Evaluate the derivatives of the following functions. \(f(x)=\sin ^{-1}\left(e^{\sin x}\right)\)

Derivatives Evaluate the derivatives of the following functions. \(f(y)=\tan ^{-1}\left(2 y^{2}-4\right)\)

Derivatives Evaluate the derivatives of the following functions. \(g(z)=\tan ^{-1}(1 / z)\)

Derivatives Evaluate the derivatives of the following functions. \(f(z)=\cot ^{-1} \sqrt{z}\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=\sec ^{-1} \sqrt{x}\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=\cos ^{-1}(1 / x)\)

Derivatives Evaluate the derivatives of the following functions. \(f(t)=\left(\cos ^{-1} t\right)^{2}\)

Derivatives Evaluate the derivatives of the following functions. \(f(u)=\csc ^{-1}(2 u+1)\)

Derivatives Evaluate the derivatives of the following functions. \(f(t)=\ln \left(\tan ^{-1} t\right)\)

Derivatives Evaluate the derivatives of the following functions. \(f(y)=\cot ^{-1}\left[1 /\left(y^{2}+1\right)\right]\)

Derivatives Evaluate the derivatives of the following functions. \(f(w)=\sin \left[\sec ^{-1}(2 w)\right]\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=\sec ^{-1}(\ln x)\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=\tan ^{-1}\left(e^{4 x}\right)\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=\csc ^{-1}\left(\tan e^{x}\right)\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=\sin \left[\tan ^{-1}(\ln x)\right]\)

Derivatives Evaluate the derivatives of the following functions. \(f(s)=\cot ^{-1}\left(e^{s}\right)\)

Derivatives Evaluate the derivatives of the following functions. \(f(x)=1 /\left(\tan ^{-1}\left(x^{2}+4\right)\right)\)

Angular size A boat sails directly toward a 150-m skyscraper that stands on the edge of a harbor. The angular size \(\theta\) of the building is the angle formed by lines from the top and bottom of the building to the observer (see figure). a. What is the rate of change of the angular size, \(d \theta / d x\), when the boat is x = 500 m from the building? b. Graph \(d \theta / d x\) as a function of x and determine the point at which the angular size changes most rapidly.

Angle of elevation A small plane flies horizontally on a line 400 m directly above an observer with a speed of 70 m/s. Let \(\theta\) be the angle of elevation of the plane (see figure). a. What is the rate of change of the angle of elevation, \(d \theta / d x\), when the plane is x = 500 m past the observer? b. Graph \(d \theta / d x\) as a function of x and determine the point at which \(\theta\) changes most rapidly.

Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\). f(x) = 3x + 4 ; (16,4)

Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\). \(f(x)=x^{2}+1\) for \(x \geq 0\); (5, 2)

Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\). f(x) = tan x; \((1, \pi / 4)\)

Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\). \(f(x)=x^{2}-2 x-3\) for \(x \leq 1 \); (12, -3)

Slopes of tangent lines Given the function f, find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\). \(f(x)=\sqrt{x} \); (2, 4)

Slopes of tangent lines Given the function f, find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\). \(f(x)=x^{3} \); (8, 2)

Slopes of tangent lines Given the function f, find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\). \(f(x)=(x+2)^{2}\); (36, 4)

Slopes of tangent lines Given the function f, find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\). \(f(x)=-x^{2}+8 \); (7, 1)

Derivatives and inverse functions Find \(\left(f^{-1}\right)^{\prime}(3)\) if \(f(x)=x^{3}+x+1\).

Derivatives and inverse functions Find the slope of the curve \(y=f^{-1}(x)\) at (4, 7) if the slope of the curve y = f(x) at (7, 4) is \(\frac{2}{3}\).

Derivatives and inverse functions If the slope of the curve \(y=f^{-1}(x)\) at (4, 7) is \(\frac{4}{5}\), find \(f^{\prime}(7)\).

Derivatives and inverse functions If the slope of the curve y = f(x) at (4, 7) is \(\frac{1}{5}\), find \(\left(f^{-1}\right)^{\prime}(7)\).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\frac{d}{d x}\left(\sin ^{-1} x+\cos ^{-1} x\right)=0\) b. \(\frac{d}{d x}\left(\tan ^{-1} x\right)=\sec ^{2} x\) c. The lines tangent to the graph of \(y=\sin ^{-1} x\) on the interval [-1, 1] have a minimum slope of 1 . d. The lines tangent to the graph of y = sin x on the interval \([-\pi / 2, \pi / 2]\) have a maximum slope of 1 . e. If f(x) = 1/x, then \(\left[f^{-1}(x)\right]^{\prime}=-\frac{1}{x^{2}}\)

Graphing f and \(f^{\prime}\) a. Graph f with a graphing utility. b. Compute and graph \(f^{\prime}\). c. Verify that the zeros of \(f^{\prime}\) correspond to points at which f has a horizontal tangent line. \(f(x)=(x-1) \sin ^{-1} x\) on [-1, 1]

Graphing f and \(f^{\prime}\) a. Graph f with a graphing utility. b. Compute and graph \(f^{\prime}\). c. Verify that the zeros of \(f^{\prime}\) correspond to points at which f has a horizontal tangent line. \(f(x)=\left(x^{2}-1\right) \sin ^{-1} x\) on [-1, 1]

Graphing f and \(f^{\prime}\) a. Graph f with a graphing utility. b. Compute and graph \(f^{\prime}\). c. Verify that the zeros of \(f^{\prime}\) correspond to points at which f has a horizontal tangent line. \(f(x)=\left(\sec ^{-1} x\right) / x\) on \([1, \infty)\)

Graphing f and \(f^{\prime}\) a. Graph f with a graphing utility. b. Compute and graph \(f^{\prime}\). c. Verify that the zeros of \(f^{\prime}\) correspond to points at which f has a horizontal tangent line. \(f(x)=e^{-x} \tan ^{-1} x\) on \([0, \infty)\)

Graphing with inverse trigonometric functions a. Graph the function \(f(x)=\frac{\tan ^{-1} x}{x^{2}+1}\). b. Compute and graph \(f^{\prime}\) and determine (perhaps approximately) the points at which \(f^{\prime}(x)=0\). c. Verify that the zeros of \(f^{\prime}\) correspond to points at which f has a horizontal tangent line.

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. f(x) = 3x - 4

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. f(x) = |x+2|, \(x \leq-2\)

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. \(f(x)=x^{2}-4\), x > 0

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. \(f(x)=\frac{x}{x+5}\)

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. \(f(x)=\sqrt{x+2}, x \geq-2\)

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. \(f(x)=x^{2 / 3}\), x > 0

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. \(f(x)=x^{-1 / 2}\), x > 0

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x$, and find the derivative of the inverse function. \(f(x)=x^{3}+3\)

Towing a boat A boat is towed toward a dock by a cable attached to a winch that stands 10 ft above the water level (see figure). Let \(\theta\) be the angle of elevation of the winch and let \(\ell\) be the length of the cable as the boat is towed toward the dock. a. Show that the rate of change of \(\theta\) with respect to \(\ell\) is \(\frac{d \theta}{d \ell}=\frac{-10}{\ell \sqrt{\ell^{2}-100}}\). b. Compute \(\frac{d \theta}{d \ell}\) when \(\ell\) = 50, 20, and 11 ft. c. Find \(\lim _{\ell \rightarrow 10^{+}} \frac{d \theta}{d \ell}\), and explain what is happening as the last foot of cable is reeled in (note that the boat is at the dock when \(\ell=10\)). d. It is evident from the figure that \(\theta\) increases as the boat is towed to the dock. Why, then, is \(d \theta / d \ell\) negative?

Tracking a dive A biologist standing at the bottom of an 80-ft vertical cliff watches a peregrine falcon dive from the top of the cliff at a \(45^{\circ}\) angle from the horizontal (see figure). a. Express the angle of elevation \(\theta\) from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80 - h.) b. What is the rate of change of \(\theta\) with respect to the bird's height when it is 60 ft above the ground?

Angle to a particle A particle travels clockwise on a circular path of diameter R, monitored by a sensor on the circle at point P; the other endpoint of the diameter on which the sensor lies is Q (see figure). Let \(\theta\) be the angle between the diameter PQ and the line from the sensor to the particle. Let c be the length of the chord from the particle's position to Q. a. Calculate \(d \theta / d c\). b. Evaluate \(\left.\frac{d \theta}{d c}\right|_{c=0}\).

Angle to a particle, part II The drawing in Exercise 59 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in the two answers.

Derivative of the inverse sine Find the derivative of the inverse sine using Theorem 3.23.

Derivative of the inverse cosine Find the derivative of the inverse cosine in the following two ways. a. Using Theorem 3.23 b. Using the identity \(\sin ^{-1} x+\cos ^{-1} x=\pi / 2\)

Derivative of \(\cot ^{-1} x\) and \(\csc ^{-1} x\) Use a trigonometric identity to show that the derivatives of the inverse cotangent and inverse cosecant differ from the derivatives of the inverse tangent and inverse secant, respectively, by a multiplicative factor of -1.

Tangents and inverses Suppose y = L(x) = ax + b (with \(a \neq 0\)) is the equation of the line tangent to the graph of a one-to-one function f at \(\left(x_{0}, y_{0}\right)\). Also, suppose that y = M(x) = cx + d is the equation of the line tangent to the graph of \(f^{-1}\) at \(\left(y_{0}, x_{0}\right)\). a. Express a and b in terms of \(x_{0}\) and \(y_{0}\). b. Express c in terms of a, and d in terms of a, \(x_{0}\), and \(y_{0}\). c. Prove that \(L^{-1}(x)=M(x)\).

Identity proofs Prove the following identities and give the values of $x$ for which they are true. \(\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}\)

Identity proofs Prove the following identities and give the values of $x$ for which they are true. \(\cos \left(2 \sin ^{-1} x\right)=1-2 x^{2}\)

Identity proofs Prove the following identities and give the values of $x$ for which they are true. \(\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}\)

Identity proofs Prove the following identities and give the values of $x$ for which they are true. \(\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}\)