91Ó°ÊÓ

Solve the given problems by finding the appropriate derivative. The insulation resistance \(R\) (in \(\Omega / \mathrm{m}\) ) of a shielded cable is given by \(R=k \ln \left(r_{2} / r_{1}\right) .\) Here \(r_{1}\) and \(r_{2}\) are the inner and outer radii of the insulation. Find the expression for \(d R / d r_{2}\) if \(k\) and \(r_{1}\) are constant.

Find the derivatives of the given functions. $$y=\frac{1}{1+4 x^{2}}-\tan ^{-1} 2 x$$

Find the derivatives of the given functions. $$s=\sin (3 \sin 2 t)$$

Solve the given problems by finding the appropriate derivative. The vapor pressure \(p\) and thermodynamic temperature \(T\) of a gas are related by the equation \(\ln p=\frac{a}{T}+b \ln T+c,\) where \(a, b\) and \(c\) are constants. Find the expression for \(d p / d T\).

Find the derivatives of the given functions. $$I=\ln \sin 2 e^{6 t}$$

Find the derivatives of the given functions. $$r=\tan (\sin 2 \pi \theta)$$

Find the derivatives of the given functions. $$r=\ln \frac{v^{2}}{v+2}$$

In Exercises \(3-36,\) evaluate each limit (if it exists). Use L'Hospital's rule (if appropriate). $$\lim _{x \rightarrow 0} \frac{\ln \sin x}{\ln \tan x}$$

Solve the given problems. A crate of weight \(w\) is being pulled along a level floor by a force \(F\) that is at an angle \(\theta\) with the floor. The force is given by \(F=\frac{0.25 w}{0.25 \sin \theta+\cos \theta} .\) Find \(\theta\) for the minimum value of \(F\)

Evaluate each limit (if it exists). Use L'Hospital's rule (if appropriate). $$\lim _{x \rightarrow 0} \frac{\ln \sin x}{\ln \tan x}$$