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\usepackage{calc}
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\usepackage{pst-plot}
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\def\ds{\displaystyle}
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\begin{document}
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\begin{center} {\large \textbf{CALCULUS SAMPLE HW DERIVATIVES}}
\end{center}
{ \textbf{Name:} \underline{\hspace{8cm}}
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\textbf{Score:} \underline{\hspace{8cm}}}
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\noindent \textbf{(1)} Find the derivative of each of the following functions. You do not need to simplify your answers.
\vspace{.5cm}
\noindent \textbf{(a)} $f(x) = \ds \frac{\cos ^8 (x)}{(e^x)^{-2}}$
\vspace{3cm}
\noindent \textbf{(b)} $f(x) = \tan (\sin ( x^2+1))$
\vspace{3cm}
\noindent \textbf{(c)} $f(x) = \sqrt{\sin ^2(x) + \cos ^2(x)}$
\vspace{3cm}
\noindent \textbf{(e)} $f(x) = 2^{\sin (x)}$
\newpage
\noindent \textbf{(2)} Find the equation of the tangent line to the graph of $f(x) = \ds \tan (e^{x^2})$ at the point where $x=0$. You may leave your answer in point-slope form.
\vspace{5cm}
\noindent \textbf{(3)} Find $y'$ (equivalently, $\ds \frac{dy}{dx}$) using implicit differentiation:
\noindent \textbf{(a)} $\sin(x - y) = xy$
\vspace{5cm}
\noindent \textbf{(b)} $e^{x+y} = y$
\newpage
\noindent \textbf{(4)} Find the slope of the tangent line to the graph of $x^2+y^2 = 5$ at the point $(1,2)$.
\vspace{7cm}
\noindent \textbf{(5)} If $f(x) = x^5$ and $g$ is its inverse function. Find $g'(y_0)$ if $y_0 = 32$.
\vspace{4cm}
\noindent \textbf{(6)} Find the derivative of $f(x) = \sin ^{-1}(4x)$.
\end{document}
