Chapter 3

Find \(d y\). $$y=\frac{2 \sqrt{x}}{3(1+\sqrt{x})}$$

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

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=e^{\left(4 \sqrt{x}+x^{2}\right)}$$

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) $$x^{2 / 3}+y^{2 / 3}=1$$

Find \(d s / d t\). $$s=\frac{\sin t}{1-\cos t}$$

Find the derivatives of the functions. $$w=(2 x-7)^{-1}(x+5)$$

Find the slope of the curve at the point indicated. $$y=\frac{x-1}{x+1}, \quad x=0$$

Find \(d y\). $$2 y^{3 / 2}+x y-x=0$$

Find the derivatives of the functions in Exercises \(23-50\). $$p=\sqrt{3-t}$$

Suppose that the dollar cost of producing \(x\) washing machines is \(c(x)=2000+100 x-0.1 x^{2}\) a. Find the average cost per machine of producing the first 100 washing machines. b. Find the marginal cost when 100 washing machines are produced. c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) $$y^{2}=e^{x^{2}}+2 x$$

Find \(d r / d \theta\). $$r=4-\theta^{2} \sin \theta$$

Find the derivatives of the functions. $$f(s)=\frac{\sqrt{s}-1}{\sqrt{s}+1}$$

Use the formula $$f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$$ to find the derivative of the functions. $$f(x)=\frac{1}{x+2}$$

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

Two commercial airplanes are flying at an altitude of \(40,000 \mathrm{ft}\) along straight-line courses that intersect at right angles. Plane \(A\) is approaching the intersection point at a speed of 442 knots (nautical miles per hour; a nautical mile is 2000 yd). Plane \(B\) is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when \(A\) is 5 nautical miles from the intersection point and \(B\) is 12 nautical miles from the intersection point?

Find \(d y\). $$x y^{2}-4 x^{3 / 2}-y=0$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\left(x^{2} \ln x\right)^{4}$$

Find the derivatives of the functions in Exercises \(23-50\). $$q=\sqrt[3]{2 r-r^{2}}$$

Suppose that the revenue from selling \(x\) washing machines is $$ r(x)=20,000\left(1-\frac{1}{x}\right) $$ dollars. a. Find the marginal revenue when 100 machines are produced. b. Use the function \(r^{\prime}(x)\) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week. c. Find the limit of \(r^{\prime}(x)\) as \(x \rightarrow \infty .\) How would you interpret this number?

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) $$y^{2}-2 x=1-2 y$$

Find \(d r / d \theta\). $$r=\theta \sin \theta+\cos \theta$$

Find the derivatives of the functions. $$u=\frac{5 x+1}{2 \sqrt{x}}$$

Use the formula $$f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$$ to find the derivative of the functions. $$f(x)=x^{2}-3 x+4$$

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

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 \(d y\). $$y=\sin (5 \sqrt{x})$$

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

Find the derivatives of the functions in Exercises \(23-50\). $$s=\frac{4}{3 \pi} \sin 3 t+\frac{4}{5 \pi} \cos 5 t$$

When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time \(t\) (hours) was \(b=10^{6}+10^{4} t-10^{3} t^{2}\). Find the growth rates at a. \(t=0\) hours. b. \(t=5\) hours. c. \(t=10\) hours.

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) $$2 \sqrt{y}=x-y$$

Find \(d r / d \theta\). $$r=\sec \theta \csc \theta$$

Find the derivatives of the functions. $$v=\frac{1+x-4 \sqrt{x}}{x}$$

Use the formula $$f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$$ to find the derivative of the functions. $$g(x)=\frac{x}{x-1}$$

The mechanics at Lincoln Automotive are reboring a 6 -in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder's radius one- thousandth of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800 in.?

Find \(d y\). $$y=\cos \left(x^{2}\right)$$

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

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

A typical male's body surface area \(S\) in square meters is often modeled by the formula \(S=\frac{1}{60} \sqrt{w h}\) where \(h\) is the height in \(\mathrm{cm},\) and \(w\) the weight in \(\mathrm{kg},\) of the person. Find the rate of change of body surface area with respect to weight for males of constant height \(\left.h=180 \mathrm{cm} \text { (roughly } 5^{\prime} 9^{\prime \prime}\right)\) Does \(S\) increase more rapidly with respect to weight at lower or higher body weights? Explain.

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) $$x y+y^{2}=1$$

Find \(d r / d \theta\). $$r=(1+\sec \theta) \sin \theta$$

Find the derivatives of the functions. $$r=2\left(\frac{1}{\sqrt{\theta}}+\sqrt{\theta}\right)$$

Use the formula $$f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$$ to find the derivative of the functions. $$g(x)=1+\sqrt{x}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\csc ^{-1}\left(x^{2}+1\right), \quad x>0$$

Sand falls from a conveyor belt at the rate of \(10 \mathrm{m}^{3} / \mathrm{min}\) onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 \(\mathrm{m}\) high? Answer in centimeters per minute.

Find \(d y\). $$y=4 \tan \left(x^{3} / 3\right)$$

It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth \(y\) of fluid in the tank \(t\) hours after the valve is opened is given by the formula $$ y=6\left(1-\frac{t}{12}\right)^{2} \mathrm{m} $$ a. Find the rate \(d y / d t(\mathrm{m} / \mathrm{h})\) at which the tank is draining at time \(t\) b. When is the fluid level in the tank falling fastest? Slowest? What are the values of \(d y / d t\) at these times? c. Graph \(y\) and \(d y / d t\) together and discuss the behavior of \(y\) in relation to the signs and values of \(d y / d t\)

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) If \(x^{3}+y^{3}=16,\) find the value of \(d^{2} y / d x^{2}\) at the point (2,2)

Find \(d p / d q\). $$p=5+\frac{1}{\cot q}$$

Find equations of all lines having slope -1 that are tangent to the curve \(y=1 /(x-1)\).