document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001098>Question 1001098</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29 ALIGN=MIDDLE  ALT=\"int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, calculate dy/dx. I know how to do the integral itself, I just don't know how to find dy/dx. Please help me solve this. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618393 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29 ALIGN=MIDDLE  ALT=\"int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;calculate dy/dx. <BR>\n" );
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document.write( "Let F(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29 ALIGN=MIDDLE  ALT=\"int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since the lower limit of integration is a function of &nbsp;x, &nbsp;the derivative of the integral is taken \r<BR>\n" );
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document.write( "with the sign &nbsp;\"minus\" &nbsp;derivative of the function representing the lower limit, &nbsp;multiplied by the value of the function \r<BR>\n" );
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document.write( "under the integral symbol taken at the current value of &nbsp;x:\r<BR>\n" );
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document.write( "F'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28d%28sin%28x%29%29%29%2F%28dx%29 ALIGN=MIDDLE  ALT=\"-%28d%28sin%28x%29%29%29%2F%28dx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283+%2B+x%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%283+%2B+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-cos%28x%29 ALIGN=MIDDLE  ALT=\"-cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283+%2B+x%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%283+%2B+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "For more clarity, &nbsp;if\r<BR>\n" );
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document.write( "F(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28f%28t%29%2Cdt%2Cg%28x%29%2Cb%29 ALIGN=MIDDLE  ALT=\"int%28f%28t%29%2Cdt%2Cg%28x%29%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "where &nbsp;b = const &nbsp;and the lower limit of integration &nbsp;g(x) &nbsp;is a function of &nbsp;x, &nbsp;then\r<BR>\n" );
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document.write( "F'(x) = -g'(x)*f(x).\r<BR>\n" );
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document.write( "For the reference, &nbsp;see &nbsp;<A  rel=nofollow HREF=https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign>this Wikipedia article</A>.\r<BR>\n" );
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