91Ó°ÊÓ

Growth of bacteria A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are \(10,000\) bacteria. At the end of 5 hours there are \(40,000 .\) How many bacteria were present initially?

In Exercises \(25-36,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\left(1-t^{2}\right) \cot h^{-1} t$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow 0} \frac{3^{x}-1}{2^{x}-1} $$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\arcsin \frac{3}{t^{2}} $$

Each of Exercises \(25-34\) gives a formula for a function \(y=f(x)\) . In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1}\) . As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.\) $$f(x)=1 / x^{3}, \quad x \neq 0$$

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\frac{1}{2} \ln \frac{1+x}{1-x} $$

find \(d y / d x.\) \begin{equation}\tan y=e^{x}+\ln x\end{equation}

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\frac{1+\ln t}{1-\ln t} $$

In Exercises \(25-36,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\cos ^{-1} x-x \operatorname{sech}^{-1} x$$

find \(d y / d x.\) \begin{equation}3+\sin y=y-x^{3}\end{equation}