3.6 The chain rule  (Page 5/6)

$y=3u-6,u=2x^2$

$y=6u^3,u=7x-4$

$y=\text{sin}u,u=5x-1$

$y=\text{cos}u,u=\frac{\text{−}x}{8}$

$y=\text{tan}u,u=9x+2$

$y=\sqrt{4u+3},u=x^2-6x$

$y={\left(3x-2\right)}^6$

$y={\left(3x^2+1\right)}^3$

$y={\text{sin}}^5\left(x\right)$

$y={\left(\frac{x}{7}+\frac{7}{x}\right)}^7$

$y=\text{tan}\left(\text{sec}x\right)$

$y=\text{csc}\left(\pi x+1\right)$

$y={\text{cot}}^{2}x$

$y=\mathrm{-6}{\text{sin}}^{\mathrm{-3}}x$

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

$y={\left(5-2x\right)}^{\mathrm{-2}}$

$y={\text{cos}}^3\left(\pi x\right)$

$y={\left(2x^3-x^2+6x+1\right)}^3$

$y=\frac{1}{{\text{sin}}^2(x)}$

$y={\left(\text{tan}x+\text{sin}x\right)}^{\mathrm{-3}}$

$y=x^2{\text{cos}}^{4}x$

$y=\text{sin}\left(\text{cos}7x\right)$

$y=\sqrt{6+\text{sec}\pi x^2}$

$y={\text{cot}}^3\left(4x+1\right)$

Let $y={\left[f(x)\right]}^3$ and suppose that $f^{\prime }\left(1\right)=4$ and $\frac{dy}{dx}=10$ for $x=1.$ Find $f\left(1\right).$

Let $y={\left(f\left(x\right)+5x^2\right)}^4$ and suppose that $f\left(\mathrm{-1}\right)=\mathrm{-4}$ and $\frac{dy}{dx}=3$ when $x=\mathrm{-1}.$ Find $f^{\prime }\left(\mathrm{-1}\right)$

Let $y={\left(f\left(u\right)+3x\right)}^2$ and $u=x^3-2x.$ If $f\left(4\right)=6$ and $\frac{dy}{dx}=18$ when $x=2,$ find $f^{\prime }\left(4\right).$

[T] Find the equation of the tangent line to $y=\text{−}\text{sin}\left(\frac{x}{2}\right)$ at the origin. Use a calculator to graph the function and the tangent line together.

[T] Find the equation of the tangent line to $y={\left(3x+\frac{1}{x}\right)}^2$ at the point $\left(1,16\right).$ Use a calculator to graph the function and the tangent line together.

Find the $x$ -coordinates at which the tangent line to $y={\left(x-\frac{6}{x}\right)}^8$ is horizontal.

[T] Find an equation of the line that is normal to $g(\theta )={\text{sin}}^2\left(\pi \theta \right)$ at the point $\left(\frac{1}{4},\frac{1}{2}\right).$ Use a calculator to graph the function and the normal line together.

$h\left(x\right)=f\left(g\left(x\right)\right);a=0$

$h\left(x\right)=g\left(f\left(x\right)\right);a=0$

$h\left(x\right)={\left(x^4+g(x)\right)}^{\mathrm{-2}};a=1$

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

$h\left(x\right)=f\left(x+f\left(x\right)\right);a=1$

$h\left(x\right)={\left(1+g(x)\right)}^3;a=2$

$h\left(x\right)=g\left(2+f\left(x^2\right)\right);a=1$

$h\left(x\right)=f\left(g\left(\text{sin}x\right)\right);a=0$

[T] The position function of a freight train is given by $s\left(t\right)=100{\left(t+1\right)}^{\mathrm{-2}},$ with $s$ in meters and $t$ in seconds. At time $t=6$ s, find the train’s

1. velocity and
2. acceleration.
3. Using a. and b. is the train speeding up or slowing down?

[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:

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

1. Determine the position of the spring at $t=1.5$ s.
2. 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 $C=0.0001x^3-0.02x^2+3x+300.$ In $t$ weeks production is estimated to be $x=1600+100t$ boxes.

1. Find the marginal cost $C^{\prime }\left(x\right).$
2. Use Leibniz’s notation for the chain rule, $\frac{dC}{dt}=\frac{dC}{dx}·\frac{dx}{dt},$ to find the rate with respect to time $t$ that the cost is changing.
3. 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.

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

1. Suppose $r=\frac{1}{{\left(t+1\right)}^2}-\frac{1}{12}$ where $t$ is time in minutes. Use the chain rule $\frac{dS}{dt}=\frac{dS}{dr}·\frac{dr}{dt}$ to find the rate at which the snowball is melting.
2. 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 $T\left(x\right)=94-10\text{cos}\left[\frac{\pi }{12}\left(x-2\right)\right],$ 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 $D\left(t\right)=5\text{sin}\left(\frac{\pi }{6}t-\frac{7\pi }{6}\right)+8,$ where $t$ is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.