3.3 Differentiation rules  (Page 7/14)

$f\left(x\right)=x^7+10$

$f\left(x\right)=5x^3-x+1$

$f\left(x\right)=4x^2-7x$

$f\left(x\right)=8x^4+9x^2-1$

$f\left(x\right)=x^4+\frac{2}{x}$

$f\left(x\right)=3x\left(18x^4+\frac{13}{x+1}\right)$

$f\left(x\right)=\left(x+2\right)\left(2x^2-3\right)$

$f\left(x\right)=x^2\left(\frac{2}{x^2}+\frac{5}{x^3}\right)$

$f\left(x\right)=\frac{x^3+2x^2-4}{3}$

$f\left(x\right)=\frac{4x^3-2x+1}{x^2}$

$f\left(x\right)=\frac{x^2+4}{x^2-4}$

$f\left(x\right)=\frac{x+9}{x^2-7x+1}$

[T] $y=3x^2+4x+1$ at $\left(0,1\right)$

[T] $y=2\sqrt{x}+1$ at $\left(4,5\right)$

[T] $y=\frac{2x}{x-1}$ at $\left(\mathrm{-1},1\right)$

[T] $y=\frac{2}{x}-\frac{3}{x^2}$ at $\left(1,\mathrm{-1}\right)$

$h\left(x\right)=4f\left(x\right)+\frac{g(x)}{7}$

$h\left(x\right)=x^{3}f(x)$

$h\left(x\right)=\frac{f\left(x\right)g(x)}{2}$

$h\left(x\right)=\frac{3f\left(x\right)}{g\left(x\right)+2}$

Find $h^{\prime }(1)$ if $h\left(x\right)=xf\left(x\right)+4g\left(x\right).$

Find $h^{\prime }\left(2\right)$ if $h\left(x\right)=\frac{f(x)}{g(x)}.$

Find $h^{\prime }\left(3\right)$ if $h\left(x\right)=2x+f\left(x\right)g\left(x\right).$

Find $h^{\prime }\left(4\right)$ if $h\left(x\right)=\frac{1}{x}+\frac{g(x)}{f(x)}.$

Let $h\left(x\right)=f\left(x\right)+g\left(x\right).$ Find

1. $h^{\prime }\left(1\right),$
2. $h^{\prime }\left(3\right),$ and
3. $h^{\prime }\left(4\right).$

Let $h\left(x\right)=f\left(x\right)g\left(x\right).$ Find

1. $h^{\prime }\left(1\right),$
2. $h^{\prime }\left(3\right),$ and
3. $h^{\prime }\left(4\right).$

Let $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}.$ Find

1. $h^{\prime }\left(1\right),$
2. $h^{\prime }\left(3\right),$ and
3. $h^{\prime }\left(4\right).$

[T] $f\left(x\right)=2x^3+3x-x^2,a=2$

[T] $f\left(x\right)=\frac{1}{x}-x^2,a=1$

[T] $f\left(x\right)=x^2-x^{12}+3x+2,a=0$

[T] $f\left(x\right)=\frac{1}{x}-x^{2\text{/}3},a=\mathrm{-1}$

Find the equation of the tangent line to the graph of $f\left(x\right)=2x^3+4x^2-5x-3$ at $x=\mathrm{-1}.$

Find the equation of the tangent line to the graph of $f\left(x\right)=x^2+\frac{4}{x}-10$ at $x=8.$

Find the equation of the tangent line to the graph of $f\left(x\right)=(3x-x^2)(3-x-x^2)$ at $x=1.$

Find the point on the graph of $f\left(x\right)=x^3$ such that the tangent line at that point has an $x$ intercept of 6.

Find the equation of the line passing through the point $P(3,3)$ and tangent to the graph of $f\left(x\right)=\frac{6}{x-1}.$

Determine all points on the graph of $f\left(x\right)=x^3+x^2-x-1$ for which the slope of the tangent line is

1. horizontal
2. $\mathrm{-1}.$

Find a quadratic polynomial such that $f\left(1\right)=5,f^{\prime }\left(1\right)=3$ and $f\text{″}\left(1\right)=\mathrm{-6}.$

A car driving along a freeway with traffic has traveled $s\left(t\right)=t^3-6t^2+9t$ meters in $t$ seconds.

1. Determine the time in seconds when the velocity of the car is 0.
2. Determine the acceleration of the car when the velocity is 0.

[T] A herring swimming along a straight line has traveled $s\left(t\right)=\frac{t^2}{t^2+2}$ feet in $t$ seconds.

Determine the velocity of the herring when it has traveled 3 seconds.

The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function $P\left(t\right)=\frac{8t+3}{0.2t^2+1},$ where $t$ is measured in years.

1. Determine the initial flounder population.
2. Determine $P^{\prime }\left(10\right)$ and briefly interpret the result.

[T] The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C\left(t\right)=\frac{2t^2+t}{t^3+50},$ where $C$ is measured in milligrams per liter of blood.

1. Find the rate of change of $C\left(t\right).$
2. Determine the rate of change for $t=8,12,24,$ and $36.$
3. Briefly describe what seems to be occurring as the number of hours increases.

A book publisher has a cost function given by $C\left(x\right)=\frac{x^3+2x+3}{x^2},$ where x is the number of copies of a book in thousands and C is the cost, per book, measured in dollars. Evaluate $C^{\prime }\left(2\right)$ and explain its meaning.

[T] According to Newton’s law of universal gravitation, the force $F$ between two bodies of constant mass $m_1$ and $m_2$ is given by the formula $F=\frac{G{m}_1{m}_2}{d^2},$ where $G$ is the gravitational constant and $d$ is the distance between the bodies.

1. Suppose that $G,m_1,\text{and}{m}_2$ are constants. Find the rate of change of force $F$ with respect to distance $d.$
2. Find the rate of change of force $F$ with gravitational constant $G=6.67\times {10}^{\mathrm{-11}}$ ${\text{Nm}}^2\text{/}{\text{kg}}^2,$ on two bodies 10 meters apart, each with a mass of 1000 kilograms.