91Ó°ÊÓ

Differentiate with respect to the independent variable. $$ f(x)=\frac{\sqrt{5 x\left(1+x^{2}\right)}}{\sqrt{2}} $$

Assume that \(f(x)\) is differentiable with respect to \(x\). Show that $$\frac{d}{d x} \ln \left[\frac{f(x)}{x}\right]=\frac{f^{\prime}(x)}{f(x)}-\frac{1}{x}$$

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(\frac{d y}{d t}\) when \(y^{2}=x^{2}-x^{4}, \frac{d x}{d t}=1\) for \(x=\frac{1}{2}\), and \(y>0\).

Bacterial Growth Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N(0) 2^{t} $$ where \(N(0)\) is the population size at time \(0 .\) Find the per capita growth rate.

Use the quotient rule to show that $$ \frac{d}{d x} \sec x=\sec x \tan x $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\frac{x-1}{x+1} $$

Differentiate with respect to the independent variable. $$ f(x)=x^{3}-\frac{1}{x^{3}} $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=2 x^{x} $$

In Problems \(63-70\), find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=x^{2} $$

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(\frac{d y}{d t}\) when \(x^{2} y=1\) and \(\frac{d x}{d t}=3\) for \(x=2\).