document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163979>Question 1163979</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Let f(x) = sin (ax+b) and let g(x) =cos(ax+b) where a and b are <BR>\n" );
document.write( "constants. Gues a formular for f^n(x) and for g^n(x) general n (positive <BR>\n" );
document.write( "integer ) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #788249 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=788249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "I'm presuming you want\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^n}{dx^n}\,\[\sin(ax\,+\,b)\]\ \forall\,n\,\in\,\mathbb{Z}^+\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\,\[\sin(ax\,+\,b)\]\ =\ a\cos(ax\,+\,b)\ =\ a\sin\(ax\,+\,b\,+\,\frac{\pi}{2}\)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2}{dx^2}\,\[\sin(ax\,+\,b)\]\ =\ \frac{d}{dx}\,\[a\sin\(ax\,+\,b\,+\,\frac{\pi}{2}\)\]\ =\ a^2\cos\(ax\,+\,b\,+\,\frac{\pi}{2})\ =\ a^2\sin\(ax\,+\,b\,+\,2\cdot\frac{\pi}{2}\)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^3}{dx^3}\,\[\sin(ax\,+\,b)\]\ =\ \frac{d}{dx}\,\[a^2\sin\(ax\,+\,b\,+\,2\cdot\frac{\pi}{2}\)\]\ =\ a^3\cos\(ax\,+\,b\,+\,2\cdot\frac{\pi}{2}\)\ =\ a^3\sin\(ax\,+\,b\,+\,3\cdot\frac{\pi}{2}\)\">\r<BR>\n" );
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document.write( "By analogy:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^n}{dx^n}\,\[\sin(ax\,+\,b)\]\ =\ a^n\sin\(ax\,+\,b\,+\,n\cdot\frac{\pi}{2}\)\">\r<BR>\n" );
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document.write( "The derivation for cosine is similar and gives the result:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^n}{dx^n}\,\[\cos(ax\,+\,b)\]\ =\ a^n\cos\(ax\,+\,b\,+\,n\cdot\frac{\pi}{2}\)\">\r<BR>\n" );
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document.write( "																<BR>\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( " \ \<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \r\"> </SPAN>\n" );
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