91Ó°ÊÓ

What is special about the functions $$f(x)=\sin ^{-1} \frac{x-1}{x+1}, \quad x \geq 0, \quad \text { and } \quad g(x)=2 \tan ^{-1} \sqrt{x} ?$$ Explain.

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt[3]{\frac{x(x-2)}{x^{2}+1}}$$

Suppose \(u\) and \(v\) are functions of \(x\) that are differentiable at \(x=0\) and that $$u(0)=5, \quad u^{\prime}(0)=-3, \quad v(0)=-1, \quad v^{\prime}(0)=2$$ Find the values of the following derivatives at \(x=0\) $$\text { a. } \frac{d}{d x}(u v)$$ $$\text { b. } \frac{d}{d x}\left(\frac{u}{v}\right)$$ $$\text { c. } \frac{d}{d x}\left(\frac{v}{u}\right)$$ $$\text { d. } \frac{d}{d x}(7 v-2 u)$$

Find the limits. $$\lim _{t \rightarrow 0} \tan \left(1-\frac{\sin t}{t}\right)$$

Does the parabola \(y=2 x^{2}-13 x+5\) have a tangent whose slope is \(-1 ?\) If so, find an equation for the line and the point of tangency. If not, why not?

How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. $$x y^{3}+x^{2} y=6$$

In Exercises \(51-70,\) find \(d y / d t\). $$y=(1+\cot (t / 2))^{-2}$$

How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. $$x^{3}+y^{2}=\sin ^{2} y$$

Find the limits. $$\lim _{\theta \rightarrow 0} \cos \left(\frac{\pi \theta}{\sin \theta}\right)$$

Suppose \(u\) and \(v\) are differentiable functions of \(x\) and that $$u(1)=2, \quad u^{\prime}(1)=0, \quad v(1)=5, \quad v^{\prime}(1)=-1$$ Find the values of the following derivatives at \(x=1\) $$\text { a. } \frac{d}{d x}(u v)$$ $$\text { b. } \frac{d}{d x}\left(\frac{u}{v}\right)$$ $$\text { c. } \frac{d}{d x}\left(\frac{v}{u}\right)$$ $$\text { d. } \frac{d}{d x}(7 v-2 u)$$.