Derivatives and Integrals An Annotated Discourse

<section xml:id="handouts" label="section-handouts">
  <title>Handouts</title>
  <introduction>
    <title>About Handouts</title>
    <p>
      Like worksheets, a <term>handout</term>
      is a division that is inteded to be printed for use in a classroom.
      In HTML output,
      you get the same print preview and page layout as you do with worksheets;
      in PDF, these will start on a new page with possibly different margins than the rest of the document.
    </p>
    <p>
      Unlike worksheets,
      handouts do not have a special class of exercises or activities
      (exercises in a handout are treated like an
      <em>inline</em> exercise).
      The other main distinction is that a handout lets workspace be specified on pretty much any block or paragraph element,
      not just exercises, tasks, and project-like elements.
      This allows them to be used in the creation of guided notes containing some premade content with lots of space for students to fill in details during class.
    </p>
  </introduction>
  <handout xml:id="handout-derivative-rules" label="handout-derivative-rules">
  <title>Derivative Rules</title>
  <page>
  <paragraphs>
    <title>Rules for specific types of functions</title>
    <dl>
      <li>
        <title>Constant functions</title>
        <p>
          <m>(k)' = 0</m>
        </p>
      </li>
      <li>
        <title>Power functions</title>
        <p>
          <m>(x^{n})' = nx^{n-1}</m>
        </p>
      </li>
      <li>
        <title>Exponential functions</title>
        <p>
          <m>(a^{x})' = \ln(a) a^{x} \text{ (for } a&gt;0\text{)}</m>
        </p>
        <p>
          <m>(e^{x})' = e^{x}</m>
        </p>
      </li>
      <li>
        <title>Logarithmic functions</title>
        <p>
          <m>(\ln(x))' = \frac{1}{x}</m>
        </p>
      </li>
      <li>
        <title>Trigonometric functions</title>
        <p>
          <m>(\sin(x))' = \cos(x)</m>.
        </p>
        <p>
          <m>(\cos(x))' = -\sin(x)</m>.
        </p>
        <p>
          <m>(\tan(x))' = \sec^{2}(x) = \frac{1}{\cos^{2}(x)}</m>.
        </p>
        <p>
          <m>(\arcsin(x))' = \frac{1}{\sqrt{1-x^{2}}}</m>.
        </p>
        <p>
          <m>(\arctan(x))' = \frac{1}{1+x^{2}}</m>.
        </p>
      </li>
    </dl>
  </paragraphs>
  <paragraphs>
    <title>Rules for combinations of functions</title>
    <dl>
      <li>
        <title>Constant multiples</title>
        <p>
          <m>(k\cdot f(x))' = k\cdot f'(x)</m>
        </p>
      </li>
      <li>
        <title>Sum and difference</title>
        <p>
          <m>(f(x) \pm g(x))' = f'(x) \pm g'(x)</m>
        </p>
      </li>
      <li>
        <title>Products (the product rule)</title>
        <p>
          <m>(f(x)\cdot g(x))' = f'(x)g(x) + f(x)g'(x)</m>
        </p>
      </li>
      <li>
        <title>Quotients (the quotient rule)</title>
        <p>
          <m>\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^{2}}</m>
        </p>
      </li>
      <li>
        <title>Compositions (the chain rule)</title>
        <p>
          <m>(f(g(x)))' = f'(g(x))\cdot g'(x)</m>
        </p>
      </li>
    </dl>
  </paragraphs>
  </page>
  </handout>
  <handout>
  <title>Guided Notes: Derivatives of sums</title>
  <page>
  <p workspace="1in">
    Today we will explore how to take the derivative of the sum of two functions.
    For example, if
    <me>
      f(x) = x^2 + 3x
    </me>,
    what is <m>f'(x)</m>?
    What two functions is this the sum of?
    What are the derivatives of each of those functions?
  </p>
  <p>
    To be sure of the derivative of the sum,
    we should use the definition of the derivative.
  </p>
  <definition workspace=".75in">
    <title>Definition of the Derivative</title>
    <statement>
      <p>
        The derivative of a function <m>f(x)</m> at any point <m>x</m> is defined as...
      </p>
    </statement>
  </definition>
  <p workspace="2in">
    Now let's apply this definition to the function <m>f(x) = x^2 + 3x</m>.
    We have:
  </p>
  <p workspace="0.25in">
    What should the general rule be?
  </p>
  </page>
  <page>
  <theorem workspace="0.5in">
    <statement>
      <p>
        For any two differentiable functions <m>f(x)</m> and <m>g(x)</m>,
        the derivative of their sum <m>h(x) = f(x) + g(x)</m> is given by:
      </p>
    </statement>
    <proof workspace="3in">
      <p>
        Let <m>f(x)</m> and <m>g(x)</m> be differentiable functions and let <m>h(x) = f(x) + g(x)</m>.
        Then by the limit definition of the derivative,
      </p>
    </proof>
  </theorem>
  <example workspace="1in">
    <statement>
      <p>
        Find the derivative of <m>f(x) = x^5 + e^x</m>.
      </p>
    </statement>
  </example>
  <example workspace="1in">
    <statement>
      <p>
        Find the derivative of <m>f(x) = \sqrt{x} + x^3 + 7x</m>.
      </p>
    </statement>
  </example>
  <example workspace="1in">
    <statement>
      <p>
        Find the derivative of <m>f(x) = 5x^4</m>.
      </p>
    </statement>
  </example>
  </page>
  </handout>
</section>