91Ó°ÊÓ

31-38. Find the indicated derivatives. If \(f(x)=12 \sqrt[3]{x^{2}}+\frac{48}{\sqrt[3]{x}}\), find \(f^{\prime}(8)\).

Velocity After \(t\) hours a passenger train is \(s(t)=24 t^{2}-2 t^{3}\) miles due west of its starting point (for \(0 \leq t \leq 12\) ). a. Find its velocity at time \(t=4\) hours. b. Find its velocity at time \(t=10\) hours. c. Find its acceleration at time \(t=1\) hour.

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(x)=\frac{3 x+1}{2+x} $$

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=3 x^{2}(2 x+1)^{5} $$

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \begin{array}{l} \text {} f(x)=a x^{2}+b x+c\\\ (a, b, \text { and } c \text { are constants) } \end{array} $$

31-38. Find the indicated derivatives. If \(f(x)=x^{3}\), find \(\left.\frac{d f}{d x}\right|_{x=-3}\)

Velocity A rocket can rise to a height of \(h(t)=t^{3}+0.5 t^{2}\) feet in \(t\) seconds. Find its velocity and acceleration 10 seconds after it is launched.

31-38. Find the indicated derivatives. If \(f(x)=x^{4}\), find \(\left.\frac{d f}{d x}\right|_{x=-2}\)

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=2 x\left(x^{3}-1\right)^{4} $$

Velocity After \(t\) hours a car is a distance \(s(t)=60 t+\frac{100}{t+3}\) miles from its starting point. Find the velocity after 2 hours.