91Ó°ÊÓ

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

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

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

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

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

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

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

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

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?