91Ó°ÊÓ

\(2-22\) Differentiate the function. \(\mathrm{H}(z)=\ln \sqrt{\frac{\mathrm{a}^{2}-z^{2}}{\mathrm{a}^{2}+z^{2}}}\)

The altitude of a triangle is increasing at a rate of 1 \(\mathrm{cm} / \mathrm{min}\) while the area of the triangle is increasing at a rate of 2 \(\mathrm{cm}^{2} / \mathrm{min}\) . At what rate is the base of the triangle changing when the altitude is 10 \(\mathrm{cm}\) and the area is 100 \(\mathrm{cm}^{2} ?\)

\(19-22\) Compute \(\Delta y\) and dy for the given values of \(x\) and \(d x=\Delta x\) . Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and \(\Delta y\) . $$ y=2 x-x^{2}, x=2, \quad \Delta x=-0.4 $$

\(2-22\) Differentiate the function. \(y=\ln \left(e^{-x}+x e^{-x}\right)\)

\(3-26\) Differentiate. $$y=\frac{v^{3}-2 v \sqrt{v}}{v}$$

\(3-32\) Differentiate the function. \(F(x)=\left(\frac{1}{2} x\right)^{5}\)

Prove the identity. $$\begin{array}{l}{(\cosh x+\sinh x)^{n}=\cosh n x+\sinh n x} \\ {(n \text { any real number) }}\end{array}$$

Prove that \(\frac{d}{d x}(\cot x)=-\csc ^{2} x\)

\(7 - 46\) Find the derivative of the function. $$y = ( 2 x - 5 ) ^ { 4 } \left( 8 x ^ { 2 } - 5 \right) ^ { - 3 }$$

(a) If \(\$ 3000\) is invested at 5\(\%\) interest, find the value of the investment at the end of 5 years if the interest is compounded ( i ) annually, (ii) semiannually, (iil) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b) If \(\mathrm{A}(\mathrm{t})\) is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by \(\mathrm{A}(\mathrm{t})\) .