Section 3.2 Product and Quotient Rules

Practice problems on differentiating functions using the product rule and quotient rule, along with related applications.

Instructions

Try each problem before opening the solution video. Each solution link opens in a new tab.

  1. Problem 1

    Differentiate the following function:

    \( \displaystyle f(x)= \sqrt[7]{x^4}(3x^2+8e^x) \)

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  2. Problem 2

    Differentiate the following function:

    \( \displaystyle g(t)=\frac{e^t}{1-2t+3t^2} \)

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  3. Problem 3

    Differentiate the following function:

    \( \displaystyle h(u)= \left(\frac{5}{u^2}+2u^3\right)7^u \)

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  4. Problem 4

    Differentiate the following function:

    \( \displaystyle f(z)= \frac{az^2}{b+ce^z} \)

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  5. Problem 5

    Find the equation of the tangent line to the curve \( \displaystyle f(x)= \frac{x}{1-2x^2} \) at the point \( (1,-1) \).

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  6. Problem 6

    If \( g(x)=xe^x \), find \( g''(1) \).

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  7. Problem 7

    Suppose that \( f(3)=-1 \), \( g(3)=2 \), \( f'(3)=-3 \), and \( g'(3)=5 \). Find \( h'(3) \) for each of the following.

    1. Part (a)

      \( h(x)=2f(x)+8g(x)+2x \)

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    2. Part (b)

      \( \displaystyle h(x)= f(x)g(x) \)

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    3. Part (c)

      \( \displaystyle h(x)= \frac{f(x)}{g(x)} \)

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    4. Part (d)

      \( \displaystyle h(x)=\frac{f(x)+x^2}{g(x)+1} \)

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