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Practice Problems for Module 6

Sections 3.3–3.6

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos. 
  1. Differentiate the functions.
    1. \(F(x)=\left(1+x+x^2\right)^{200}\)
    2. \(g(\theta)=\cos^2\theta\)

      1. \(F'(x)=200\left(1+x+x^2\right)^{199}(1+2x)\)
      2. \(g'(\theta)=-2\cos\theta \sin\theta\)

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      MLC WIR 20B M151 week4 #1kl


    3. \(y=e^{\tan \theta}\)
    4. \(F(t)=e^{t\sin 2t}\)

      1. \(y'=e^{\tan\theta}\sec^2\theta\)
      2. \(F'(t)=e^{t\sin(2t)}\left(\sin(2t)+2t\cos(2t)\right)\)

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      MLC WIR 20B M151 week4 #1mn


    5. \(f(t)=\tan (\sec (\cos t))\)

      \(f'(t)=\sec^2(\sec(\cos t))\sec(\cos t)\tan(\cos t)(-\sin t)\)

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      MLC WIR 20B M151 week4 #1o


  2. Find \(y'\) and \(y''\) for \(y=e^{e^x}\)

    \(y'=e^{e^x}e^x\)
    \(y''=e^{e^x+x}\left(e^x+1\right)\)

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    MLC WIR 20B M151 week4 #2b


  3. Let \(r(x)=f(g(h(x)))\), where \(h(1)=2\), \(g(2)=3\), \(h'(1)=4\), \(g'(2)=5\), and \(f'(3)=6\). Find \(r'(1)\).

    \(r'(1)=120\)

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    MLC WIR 20B M151 week4 #3


  4. If \(g(x)=f(3f(4f(x)))\), where \(f(0)=0\) and \(f'(0)=2\), find \(g'(0)\).

    \(g'(0)=96\)

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    MLC WIR 20B M151 week4 #4


  5. Find the \(2020\)th derivative of \(y=\cos 2x\).

    \(y^{(2020)}=2^{2020}\cos(2x)\)

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    MLC WIR 20B M151 week4 #5


  6. Find the \(2020\)th derivative of \(f(x)=xe^{-x}\).

    \(f^{(2020)}(x)=-(2020-x)e^{-x}\)

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    MLC WIR 20B M151 week4 #6


  7. If \(f\) and \(g\) are the functions whose graphs are shown, let \(u(x)=f(g(x))\), \(v(x)=g(f(x))\), and \(w(x)=g(g(x))\). Find \(u'(1)\), \(v'(1)\), and \(w'(1)\).
    Screen-Shot-2022-01-27-at-4-42-41-PM.png

    \(u'(1)=\dfrac{3}{4}\)
    \(v'(1)=DNE\)
    \(w'(1)=-2\)

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    MLC WIR 20B M151 week4 #7


  8. Find \(\dfrac{dy}{dx}\).
    1. \(x^3-xy^2+y^3=1\).

      \( \dfrac{dy}{dx}=\dfrac{-3x^2+y^2}{-2xy+3y^2}\)

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      MLC WIR 20B M151 week5 #1a


    2. \(\cos(xy)=1+\sin y\).

      \( \dfrac{dy}{dx}=\dfrac{-y\sin(xy)}{\cos y +x\sin(xy)}\)

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      MLC WIR 20B M151 week5 #1b


    3. \(e^y\sin x = x + xy\).

      \( \dfrac{dy}{dx}=\dfrac{1+y-e^y\cos x}{e^y \sin x -x}\)

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      MLC WIR 20B M151 week5 #1c


    4. \(x\sin y + y \sin x = 1\).

      \( \dfrac{dy}{dx}=\dfrac{-\sin y - y \cos x}{x\cos y + \sin x}\)

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      MLC WIR 20B M151 week5 #1d


  9. ​If \(g(x)+x \sin ( g(x))=x^2\), find \(g'(x)\).

    \(g'(x)=\dfrac{2x-\sin (g(x))}{1+x\cos(g(x))}\)

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    MLC WIR 20B M151 week5 #2


  10. Find an equation of the tangent line to the curve at the given point.
    1. \(x^2+2xy+4y^2=12, \quad (2,1)\)

      \(y=-\dfrac{1}{2}x+2\)

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      MLC WIR 20B M151 week5 #3a


    2. \(y\sin 2x = x\cos 2y, \quad \left(\dfrac{\pi}{2},\dfrac{\pi}{4}\right)\)

      \(y=\dfrac{1}{2}x\)

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      MLC WIR 20B M151 week5 #3b


  11. ​Find the derivative of the function:
    1. \(y=\left(\tan^{-1} x\right)^2\)
    2. \(y=\tan^{-1}\left(x^2\right)\)​

      1. \(\dfrac{dy}{dx}=2\left(\tan^{-1} x\right) \dfrac{1}{1+x^2}\)
      2. \(\dfrac{dy}{dx}=\dfrac{2x}{1+x^4}\)

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      MLC WIR 20B M151 week5 #4ab


    3. \(R(t)=\arcsin\left(\dfrac{1}{t}\right)\)
    4. \(f(x)=\arctan \left(x^2-x\right)\)

      1. \(\dfrac{dR}{dt}=\dfrac{-1}{t^2\sqrt{1-\dfrac{1}{t^2}}}\)
      2. \(\dfrac{df}{dx}=\dfrac{2x-1}{1+\left(x^2-x\right)^2}\)

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      MLC WIR 20B M151 week5 #4cd


    5. \(f(x)=\ln (\sin^2 x)\)
    6. \(g(x)=\ln \left(xe^{-2x}\right)\)

      1. \(f'(x)=\dfrac{2\cos x}{\sin x}\)
      2. \(g'(x)=\dfrac{1-2x}{x}\)

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      MLC WIR 20B M151 week5 #4ef


    7. \(f(x)=\log(1+\cos x)\)
    8. \(F(s)=\ln \ln s\)
    9. \(y=\log_2\left(x\log_5 x\right)\)

      1. \(f'(x)=\dfrac{-\sin x}{(1+\cos x)\ln 10}\)
      2. \(F'(s)=\dfrac{1}{s\ln s}\)
      3. \(y'=\dfrac{\log_5 x + \dfrac{1}{\ln 5}}{\left(x\log_5 x\right)\ln 2}\)

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      MLC WIR 20B M151 week5 #4ghi


  12. If \(f(x)=\cos \left(\ln x^2\right)\)

    \(f'(1)=0\)

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    MLC WIR 20B M151 week5 #5


  13. Find the equation of the tangent line to the curve \(y=x^2\ln x\) at the point \((1,0)\).

    \(y=x-1\)

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    MLC WIR 20B M151 week5 #6


  14. Use the logarithmic differentiation to find the derivative of the function.
    1. \( y=\dfrac{e^{-x}\cos^2x}{x^2+x+1}\).

      \(y'=\dfrac{e^{-x}\cos^2x}{x^2+x+1}\left(-1-\dfrac{2\sin x}{\cos x} - \dfrac{2x+1}{x^2+x+1}\right)\)

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      MLC WIR 20B M151 week5 #7a


    2. \( y=x^x\)

      \(y'=x^2(\ln x + 1)\)

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      MLC WIR 20B M151 week5 #7b


    3. \( y=(\ln x)^{\cos x}\)

      \(y'=(\ln x)^{\cos x} \left( -\sin x \cdot \ln (\ln x) + \dfrac{\cos x}{x\ln x}\right)\)

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      MLC WIR 20B M151 week5 #7c