Derivative-rules-exponential-and-logarithmic-functions.tex

\Section{Derivative rules---exponential and logarithmic functions}
\begin{multicols}{2}
 \begin{FormulaBox}{Derivative of $\ln$}
  Derivative of natural logarithm is reciprocal:
  \WithSymbolDefs{
   \[ \D{\big.\LN{\X}} = \frac{1}{\X} \]
  }
  \bigskip
  With chain rule:
  \WithSymbolDefs{
   \[
    \D{\big.\LN{\W}}
    = \pfrac{1}{\W} \cdot \Deriv{\W}{\X}
   \]
  }
  \WithWordDefs{
   \[
    \D{\big.\LN{\W}}
    = \pfrac{1}{\W} \cdot \D{\W}
   \]
  }
 \end{FormulaBox}
 \begin{FormulaBox}{Derivative of $\me^x$}
  Natural exponential function is its own derivative:
  \WithSymbolDefs{
   \[ \D{\big.\me^{\X}} = \me^{\X} \]
  }
  \bigskip
  With chain rule:
  \WithSymbolDefs{
   \[
    \D{\big.\me^{\W}}
    = \me^{\W} \cdot \D{\W}
   \]
  }
  \WithWordDefs{
   \[
    \D{\me^{\W}}
    = \me^{\W} \cdot \D{\W}
   \]
  }
 \end{FormulaBox}
 \begin{FormulaBox}{Derivative of other logarithms}
  For a different base, convert to $\ln$ first.
  \begin{equation*}
   \log_b (x) = \frac{\LN{x}}{\LN{b}}
  \end{equation*}
  \WithSymbolDefs{
   \[
    \D{\big.\log_b(\X)} = \frac{1}{\LN{b} \cdot \X}
   \]
  }
 \end{FormulaBox}
 \begin{FormulaBox}{Derivative of other exponentials}
  For a different base, convert to $\me$ first.
  \begin{equation*}
   b^x = \me^{x \cdot \LN{b}}
  \end{equation*}
  \WithSymbolDefs{
   \[
    \D{b^\X} = \LN{b} \cdot \me^{\X \cdot \LN{b}} = \LN{b} \cdot b^\X
   \]
  }
 \end{FormulaBox}
\end{multicols}

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