document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=355601>Question 355601</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation:\r<BR>\n" );
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document.write( "a) Differentiate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+cos%5E%28-1%29+%28e%5Ex%29%2F+root%283%2C+x%29+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+cos%5E%28-1%29+%28e%5Ex%29%2F+root%283%2C+x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > DO NOT SIMPLIFY<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) Simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin+%28+tan%5E%28-1%29+%282x%29+%29+ ALIGN=MIDDLE  ALT=\"+sin+%28+tan%5E%28-1%29+%282x%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #253977 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=253977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+cos%5E%28-1%29+%28e%5Ex%29%2F+root%283%2C+x%29+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+cos%5E%28-1%29+%28e%5Ex%29%2F+root%283%2C+x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This problem uses the several levels of the chain rule. First we will<BR>\n" );
document.write( "Let u = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E%28-1%29+%28e%5Ex%29 ALIGN=MIDDLE  ALT=\"cos%5E%28-1%29+%28e%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+root%283%2C+x%29+=+x%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"v+=+root%283%2C+x%29+=+x%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "This makes f = u/v and from this we know that<BR>\n" );
document.write( "f' = (v*u' - u*v')/v^2<BR>\n" );
document.write( "For this we will need u' and v':<BR>\n" );
document.write( "u' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fsqrt%281+-+%28e%5Ex%29%5E2%29%29%2A%28d%2Fdx%29%28e%5Ex%29 ALIGN=MIDDLE  ALT=\"%281%2Fsqrt%281+-+%28e%5Ex%29%5E2%29%29%2A%28d%2Fdx%29%28e%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > this becomes:<BR>\n" );
document.write( "u' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fsqrt%281-e%5E%282x%29%29%29%2Ae%5Ex+=+e%5Ex%2Fsqrt%281-e%5E%282x%29%29 ALIGN=MIDDLE  ALT=\"%281%2Fsqrt%281-e%5E%282x%29%29%29%2Ae%5Ex+=+e%5Ex%2Fsqrt%281-e%5E%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and v' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29x%5E%281-%281%2F3%29%29+=+%281%2F3%29x%5E%28-2%2F3%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29x%5E%281-%281%2F3%29%29+=+%281%2F3%29x%5E%28-2%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting u, u', v and v' into the f' equation we get:<BR>\n" );
document.write( "f' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E%281%2F3%29%29%2A%28e%5Ex%2Fsqrt%281-e%5E%282x%29%29%29+-+%28cos%5E%28-1%29+%28e%5Ex%29%29%2A%28%281%2F3%29x%5E%28-2%2F3%29%29%29%2F%28x%5E%281%2F3%29%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And if you're not supposed to simplify, then I guess this mess is an acceptable answer.<br><BR>\n" );
document.write( "b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin+%28+tan%5E%28-1%29+%282x%29+%29+ ALIGN=MIDDLE  ALT=\"+sin+%28+tan%5E%28-1%29+%282x%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For this one, picture a right triangle. For one of the acute angles we want the tangent to be 2x. In other words we want the ratio of opposite/adjacent to be 2x. An opposite side of 2x and an adjacent side of 1 would give us this ratio. We want to find the sin of this angle. Since sin is opposite/hypotenuse, we will need an expression for the hypotenuse. For this we can use the Pythagorean Theorem: <BR>\n" );
document.write( "Opposite^2 + Adjacent^2 = Hypotenuse^2<BR>\n" );
document.write( "Putting our expressions for the opposite side and adjacent side into this we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%29%5E2+%2B+%281%29%5E2 ALIGN=MIDDLE  ALT=\"%282x%29%5E2+%2B+%281%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = Hypotenuse^2<BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > = Hypotenuse^2<BR>\n" );
document.write( "So the Hypotenuse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"sqrt%284x%5E2+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can express the sin ratio:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin+%28+tan%5E%28-1%29+%282x%29+%29+=+%282x%29%2Fsqrt%284x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"+sin+%28+tan%5E%28-1%29+%282x%29+%29+=+%282x%29%2Fsqrt%284x%5E2+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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