91Ó°ÊÓ

Find \(d y / d u, d u / d x,\) and \(d y / d x.\) $$ y=u^{-1}, u=x^{3}+2 x^{2} $$

Moving Point A point is moving along the graph of \(y=x^{2}\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x .\) $$ \text { (a) } x=-3 \quad \text { (b) } x=0 \quad \text { (c) } x=1 \quad \text { (d) } x=3 $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(y+x y=4 \quad(-5,-1)\)

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$\begin{array}{ll}{\text { Function }} & {\text { Point }} \\\\{f(t)=\frac{t^{2}-1}{t+4}} & {(1,0)} \end{array}$$

Medicine The effectiveness \(E\) (on a scale from 0 to 1 ) of a pain-killing drug \(t\) hours after entering the bloodstream is given by $$ E=\frac{1}{27}\left(9 t+3 t^{2}-t^{3}\right), \quad 0 \leq t \leq 4.5 $$ Find the average rate of change of \(E\) on each indicated interval and compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ \begin{array}{llll}{\text { (a) }[0,1]} & {\text { (b) }[1,2]} & {\text { (c) }[2,3]} & {\text { (d) }[3,4]}\end{array} $$

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6-2 x ;(2,2) $$

Find the derivative of the function. $$ s(t)=t^{3}-2 t+4 $$

find the second derivative of the function. $$ y=x^{2}\left(x^{2}+4 x+8\right) $$

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \(x^{3}-y^{2}=0\) \(\quad\) \((1,1)\)

Moving Point A point is moving along the graph of \(y=1 /\left(1+x^{2}\right)\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x .\) $$ \text { (a) } x=-2 \quad \text { (b) } x=2 \quad \text { (c) } x=0 \quad \text { (d) } x=10 $$