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\problemgrouptitle{Higher Derivatives}

\begin{problemblock}[2]{Calculate the second derivative for each of the following functions.}

\item $f(x)=\cot x$
\answer{$\displaystyle \frac{d^2f}{dx^2}=2\csc^2 x\cot x$}

\item $f(x)=(x-2)^{10}$ ; Also find $f''(3)$~.
\answer{$f''(x)=90(x-2)^{8}$ ; $f''(3)=90$}

\item $y=x^3\sec x$
\answer{$y''=6x\sec x+6x^2\sec x\tan x+x^3\sec x\tan^2x+x^3\sec^3x$}

\item $x^2-y^2=16$ (Use implicit differentiation.)
\answer{$\displaystyle y''=\frac{y-xy'}{y^2}=\frac{1}{y}-\frac{x^2}{y^3}$}

\end{problemblock}


\begin{problemblock}{When one uses derivatives, their simplification becomes important.}

\item Show that $f''(x)=\displaystyle\frac{32\left(3x^2+16\right)}{\left(x^2-16\right)^3}$ for $f(x)=\displaystyle\frac{x^2}{x^2-16}$.

\item Show that $f''(x)=\displaystyle\frac{8x+8}{(x-2)^4}$ for $f(x)=\displaystyle\frac{x^2}{x^2-4x+4}$.

\end{problemblock}

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