합성함수의 미분법

[T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where $t$ is measured in seconds and $s$ is in inches:

$s(t)=-3\cos \left(\pi t+\frac{\pi }{4}\right).$

a. Determine the position of the spring at $t=1.5$ s. b. Find the velocity of the spring at $t=1.5$ s.

[T] The total cost to produce $x$ boxes of Thin Mint Girl Scout cookies is $C$ dollars, where $C=0.0001{x}^{3}-0.02{x}^{2}+3x+300.$ In $t$ weeks production is estimated to be $x=1600+100t$ boxes.

a. Find the marginal cost ${C}^{'}(x).$ b. Use Leibniz’s notation for the chain rule, $\frac{dC}{dt}=\frac{dC}{dx}\cdot \frac{dx}{dt},$ to find the rate with respect to time $t$ that the cost is changing. c. Use b. to determine how fast costs are increasing when $t=2$ weeks. Include units with the answer.

[T] The formula for the area of a circle is $A=\pi {r}^{2},$ where $r$ is the radius of the circle. Suppose a circle is expanding, meaning that both the area $A$ and the radius $r$ (in inches) are expanding.

a. Suppose $r=2-\frac{100}{{(t+7)}^{2}}$ where $t$ is time in seconds. Use the chain rule $\frac{dA}{dt}=\frac{dA}{dr}\cdot \frac{dr}{dt}$ to find the rate at which the area is expanding. b. Use a. to find the rate at which the area is expanding at $t=4$ s.

[T] The formula for the volume of a sphere is $S=\frac{4}{3}\pi {r}^{3},$ where $r$ (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

a. Suppose $r=\frac{1}{{(t+1)}^{2}}-\frac{1}{12}$ where $t$ is time in minutes. Use the chain rule $\frac{dS}{dt}=\frac{dS}{dr}\cdot \frac{dr}{dt}$ to find the rate at which the snowball is melting. b. Use a. to find the rate at which the volume is changing at $t=1$ min.

[T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function $T(x)=94-10\cos \left[\frac{\pi }{12}(x-2)\right],$ where $x$ is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.

[T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function $D(t)=5\sin \left(\frac{\pi }{6}t-\frac{7\pi }{6}\right)+8,$ where $t$ is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

T/F: The Chain Rule describes how to evaluate the derivative of a composition of functions.

T/F: The Generalized Power Rule states that $\displaystyle \frac{d}{dx}\Big( g(x)^n\Big) = n\big(g(x)\big)^{n-1}$.

T/F: $\displaystyle \frac{d}{dx}\big(\ln (x^2)\big) = \frac{1}{x^2}$.

T/F: $\displaystyle \frac{d}{dx}\big(3^x\big) \approx 1.1\cdot3^x$.

T/F: $\displaystyle \frac{dx}{dy} = \frac{dx}{dt}\cdot \frac{dt}{dy}$

$f(x) = \big(\ln x + x^2\big)^3$

In Exercises 7–36, compute the derivative of the given function.

$f(x) = (4x^3-x)^{10}$

In Exercises 7–36, compute the derivative of the given function.

$f(t) = (3t-2)^{5}$

In Exercises 7–36, compute the derivative of the given function.

$g(\theta) = (\sin \theta+\cos \theta)^3$

In Exercises 7–36, compute the derivative of the given function.

$h(t) = e^{3t^2+t-1}$

In Exercises 7–36, compute the derivative of the given function.

$f(x) = 2^{x^3+3x}$

In Exercises 7–36, compute the derivative of the given function.

$f(x) = \big(x+\frac{1}{x}\big)^4$

In Exercises 7–36, compute the derivative of the given function.

$f(x) = \cos (3x)$

In Exercises 7–36, compute the derivative of the given function.

$g(x) = \tan (5x)$