91Ó°ÊÓ

Differentiate with respect to the independent variable. $$ g(s)=\frac{s^{1 / 3}-1}{s^{2 / 3}-1} $$

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

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} 2 x & \text { for } x \leq 1 \\ x+2 & \text { for } x>1 \end{array}\right. $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x^{3}-x^{2} $$

Assume that the radius \(r\) and the surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t .\) Express \(d S / d t\) in terms of \(d r / d t\)

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the radioactive decay rate of the material is \(4 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

Find the derivative of $$ f(x)=\sin \sqrt{3 x^{3}+3 x} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the half-life of the material is 3 days. Find the differential equation for the radioactive decay function \(W(t)\).

Differentiate with respect to the independent variable. $$ g(s)=\frac{s^{1 / 7}-s^{2 / 7}}{s^{3 / 7}+s^{4 / 7}} $$

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