91Ó°ÊÓ

Find the first and second derivatives of the function. \(g(t)=2 \cos t-3 \sin t\)

\(7-42=\) Find the derivative of the function. $$y=\left(\frac{1-\cos 2 x}{1+\cos 2 x}\right)^{4}$$

Each limit represents the derivative of some function \(f\) at some number \(a .\) State such an \(f\) and \(a\) in each case. $$ \lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5} $$

If $$f(x)=\sec x, \text { find } f^{\prime \prime}(\pi / 4)$$

Each limit represents the derivative of some function \(f\) at some number \(a .\) State such an \(f\) and \(a\) in each case. $$ \lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4} $$

Show by implicit differentiation that the tangent to the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ at the point \(\left(x_{0}, y_{0}\right)\) is $$\frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}=1$$

When air expands adiabatically (without gaining or losing heat), its pressure \(P\) and volume \(V\) are related by the equation \(P V^{1.4}=C,\) where \(C\) is a constant. Suppose that at a certain instant the volume is 400 \(\mathrm{cm}^{3}\) and the pressure is 80 \(\mathrm{kPa}\) and is decreasing at a rate of 10 \(\mathrm{kPa} / \mathrm{min.}\) At what rate is the volume increasing at this instant?

\(7-42=\) Find the derivative of the function. $$y=\left(a x+\sqrt{x^{2}+b^{2}}\right)^{-2}$$

Find the first and second derivatives of the function. \(h(t)=\sqrt{t}+5 \sin t\)

If $$ H(\theta)=\theta \sin \theta, \text { find } H^{\prime}(\theta) \text { and } H^{\prime \prime}(\theta)$$