미분법 전체 문제

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={({x}^{2}-1)}^{\ln x}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{\cot x}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y=\frac{x+11}{\sqrt[3]{{x}^{2}-4}}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{-1/2}{({x}^{2}+3)}^{2/3}{(3x-4)}^{4}$

[T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1964 is modeled by the function

$N(t)=5.3{e}^{0.093{t}^{2}-0.87t},(0\le t\le 4),$ where $N(t)$ gives the number of cases (in thousands) and t is measured in years, with $t=0$ corresponding to the beginning of 1960.

a. Show work that evaluates $N(0)$ and $N(4).$ Briefly describe what these values indicate about the disease in New York City. b. Show work that evaluates ${N}^{'}(0)$ and ${N}^{'}(3).$ Briefly describe what these values indicate about the disease in New York City.

[T] Using a computer program or a calculator, fit a growth curve to the data of the form $p=a{b}^{t}.$

For the following exercises, find the quantities for the given equation.

Find $\frac{dy}{dt}$ at $x=1$ and $y={x}^{2}+3$ if $\frac{dx}{dt}=4.$

For the following exercises, find the quantities for the given equation.

Find $\frac{dx}{dt}$ at $x=-2$ and $y=2{x}^{2}+1$ if $\frac{dy}{dt}=-1.$

For the following exercises, find the quantities for the given equation.

Find $\frac{dz}{dt}$ at $(x,y)=(1,3)$ and ${z}^{2}={x}^{2}+{y}^{2}$ if $\frac{dx}{dt}=4$ and $\frac{dy}{dt}=3.$

[T] If two electrical resistors are connected in parallel, the total resistance (measured in ohms, denoted by the Greek capital letter omega, $\Omega )$ is given by the equation $\frac{1}{R}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}.$ If ${R}_{1}$ is increasing at a rate of $0.5\,\Omega /\min$ and ${R}_{2}$ decreases at a rate of $1.1\text{Ω/min},$ at what rate does the total resistance change when ${R}_{1}=20\Omega$ and ${R}_{2}=50\Omega$?

You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16 mph east and your friend riding $12\,\text{mph}$ north. After you traveled $4\,\text{mi,}$ at what rate is the distance between you changing?

A 5-ft-tall person walks toward a wall at a rate of 2 ft/sec. A spotlight is located on the ground 40 ft from the wall. How fast does the height of the person’s shadow on the wall change when the person is 10 ft from the wall?

A helicopter starting on the ground is rising directly into the air at a rate of 25 ft/sec. You are running on the ground starting directly under the helicopter at a rate of 10 ft/sec. Find the rate of change of the distance between the helicopter and yourself after 5 sec.

You are stationary on the ground and are watching a bird fly horizontally at a rate of $10$ m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m?

Using the same setup as the previous problem, determine at what rate the beam of light moves across the beach 1 mi away from the closest point on the beach.

[T] A batter hits a ball toward third base at 75 ft/sec and runs toward first base at a rate of 24 ft/sec. At what rate does the distance between the ball and the batter change when 2 sec have passed?

In Exercises 15–36, compute the derivative of the given function.

$f(x) = x\sin x$

In Exercises 15–36, compute the derivative of the given function.

$f(x) = x^2\cos x$

In Exercises 15–36, compute the derivative of the given function.

$f(x) = e^x\ln x$

In Exercises 15–36, compute the derivative of the given function.

$\displaystyle f(t) = \frac{1}{t^2}(\csc t-4)$