Derivatives and Integrals An Annotated Discourse

<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>