11-lesson.tex

\section*{Derivative Review. Chain Rule}

Basic Formulas
\begin{equation*}
  \begin{aligned}
    \frac{d}{dx} f(g(x)) &= \frac{df}{dx}(g(x)))\frac{d}{dx} g(x) \\
    \noalign{\vspace{1 em}or if}
    h(x) &= f(g(x)) \\
    \noalign{then}
    h'(x) &= f^\prime(g(x))g'(x) \\
  \end{aligned}
\end{equation*}

\begin{questions}
\question
Calculate the derivative of $f(x)=\ln (x^3)$ two ways. Use the logarithm property and chain rule
\begin{solution}[1.5in]
  Logarithms
  $$f(x) = 3 \ln(x)$$
  so
  $$f^\prime(x)=\frac{3}{x}$$
  Chain Rule
  $$f(x)=g(h(x))$$
  where   $g(x)=\ln(x)$ and $h(x)=x^3$
  $$g'(x) = \frac{1}{x}$$
  $$h'(x) = 3x^2$$
  so
  $$f'(x)=g'(h(x))h'(x) = \frac{1}{x^3}\cdot 3x^2 = \frac{3x^2}{x^3} = \frac{3}{x}$$
\end{solution}
\question
Calculate the derviative of $e^{x^2}$
\begin{solution}[1.5in]
  $$f(x)=e^{x}$$
  $$g(x)=x^2$$
  $$f'(x)=e^{x}$$
  $$g'(x)=2x$$
  so
  $$h'(x) = 2xe^{x^2}$$
\end{solution}
\question
Given the table
  \begin{tabular}{c|cccccc}
    \hline
    & $f(x)$ & $f^\prime(x)$ & $g(x)$ & $g'(x)$ \\
    \hline
    1& 1 & 3 & 5 & 4 \\
    \hline
    3& 9 & 7 & -2 & 3 \\
    \hline
    5 &3 & 6 & 7 & 5 \\
    \hline
  \end{tabular}
  \vspace{1em}
  \begin{parts}
    \part Calculate $h'(1)$ where $h(x) = f(g(x))$
    \begin{solution}[.5in]
      $h'(x)=f'(g(x))g'(x)$ so $h'(1)= f'(5)g'(1)=6\cdot 4=24$
    \end{solution}
    \part Calculate $h'(5)$ where $h(x) = g(f(x))$.
    \begin{solution}[.5in]
      $h'(x)=g'(f(x))f'(x)$ so $h'(5)= g'(3)f'(5)=3\cdot 6=18$
    \end{solution}
  \end{parts}
\end{questions}