91Ó°ÊÓ

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{2}{x^{2}}-\frac{3}{x^{1 / 3}}\)

At a temperature of \(20^{\circ} \mathrm{C}\), the volume \(V\) (in liters) of \(1.33 \mathrm{~g}\) of \(\mathrm{O}_{2}\) is related to its pressure \(p\) (in atmospheres) by the formula \(V=1 / p\) a. What is the average rate of change of \(V\) with respect to \(p\) as \(p\) increases from \(p=2\) to \(p=3 ?\) b. What is the rate of change of \(V\) with respect to \(p\) when \(p=2 ?\)

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{-}} \frac{1+x}{1-x}\)

Find the indicated limit. \(\lim _{x \rightarrow 2} \frac{2 x+1}{x+2}\)

Suppose \(f\) and \(g\) are functions that are differentiable at \(x=1\) and that \(f(1)=2, f^{\prime}(1)=-1\), \(g(1)=-2\), and \(g^{\prime}(1)=3 .\) Find the value of \(h^{\prime}(1)\) \(h(x)=\frac{x f(x)}{x+g(x)}\)

Find the derivative of each function. \(f(x)=\left(\frac{x+1}{x-1}\right)^{5}\)

Suppose \(f\) and \(g\) are functions that are differentiable at \(x=1\) and that \(f(1)=2, f^{\prime}(1)=-1\), \(g(1)=-2\), and \(g^{\prime}(1)=3 .\) Find the value of \(h^{\prime}(1)\) \(h(x)=\frac{f(x) g(x)}{f(x)-g(x)}\)

The total cost \(C(x)\) (in dollars) incurred by Aloha Company in manufacturing \(x\) surfboards a day is given by $$ C(x)=-10 x^{2}+300 x+130 \quad(0 \leq x \leq 15) $$ a. Find \(C^{\prime}(x)\). b. What is the rate of change of the total cost when the level of production is ten surfboards a day?

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{3}{x^{3}}+\frac{4}{\sqrt{x}}+1\)

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{+}} \frac{1+x}{1-x}\)