91Ó°ÊÓ

\(23-26 \text { Find } y^{\prime} \text { and } y^{\prime \prime}\) \(y=\frac{\ln x}{x^{2}}\)

\(3-26\) Differentiate. $$f(x)=\frac{x}{x+\frac{c}{x}}$$

A water trough is 10 \(\mathrm{m}\) long and a cross-section has the shape of an isosceles trapezoid that is 30 \(\mathrm{cm}\) wide at the bottom, 80 \(\mathrm{cm}\) wide at the top, and has height 50 \(\mathrm{cm} .\) If the trough is being filled with water at the rate of 0.2 \(\mathrm{m}^{3} / \mathrm{min}\) , how fast is the water level rising when the water is 30 \(\mathrm{cm}\) deep?

(a) Find an equation of the tangent line to the curve \(y=2 x \sin x\) at the point \((\pi / 2, \pi) .\) (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

\(23-28\) Use a linear approximation (or differentials) to estimate the given number. \((8.06)^{2 / 3}\)

\(23-26 \text { Find } y^{\prime} \text { and } y^{\prime \prime}\) \(y=\ln \left(x+\sqrt{1+x^{2}}\right)\)

\(25-30\) Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2}+x y+y^{2}=3, \quad(1,1) \quad\) (ellipse)

\(3-32\) Differentiate the function. \(y=4 \pi^{2}\)

\(7 - 46\) Find the derivative of the function. $$F ( z ) = \sqrt { \frac { z - 1 } { z + 1 } }$$

(a) Find an equation of the tangent line to the curve \(y=\sec x-2 \cos x\) at the point \((\pi / 3,1) .\) (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.