document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204864>Question 1204864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if : f '(g (x)) = g '(x), f ''(x) \[Times] f '(x) = f (x), g '(3) = 2 g ''(3) = 2} then (d ^5 g)/(dx ^5)| = ... (9 , 10 ,18 , 24) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847975 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=847975\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Given:**\r<BR>\n" );
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document.write( "* f'(g(x)) = g'(x) <BR>\n" );
document.write( "* f''(x) * f'(x) = f(x)<BR>\n" );
document.write( "* g'(3) = 2<BR>\n" );
document.write( "* g''(3) = 2\r<BR>\n" );
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document.write( "**2. Find g'''(3):**\r<BR>\n" );
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document.write( "* Differentiate f'(g(x)) = g'(x) with respect to x using the chain rule:<BR>\n" );
document.write( "   f''(g(x)) * g'(x) = g''(x)\r<BR>\n" );
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document.write( "* Substitute x = 3:<BR>\n" );
document.write( "   f''(g(3)) * g'(3) = g''(3)<BR>\n" );
document.write( "   f''(g(3)) * 2 = 2 <BR>\n" );
document.write( "   f''(g(3)) = 1\r<BR>\n" );
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document.write( "**3. Find g''''(3):**\r<BR>\n" );
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document.write( "* Differentiate f''(g(x)) * g'(x) = g''(x) with respect to x using the product rule and chain rule:<BR>\n" );
document.write( "   [f'''(g(x)) * g'(x)] * g'(x) + f''(g(x)) * g''(x) = g'''(x)\r<BR>\n" );
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document.write( "* Substitute x = 3:<BR>\n" );
document.write( "   [f'''(g(3)) * 2] * 2 + 1 * 2 = g'''(3)<BR>\n" );
document.write( "   4 * f'''(g(3)) + 2 = g'''(3) \r<BR>\n" );
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document.write( "* We need to find f'''(g(3)). To do this, differentiate the given equation f''(x) * f'(x) = f(x) with respect to x using the product rule:<BR>\n" );
document.write( "   f'''(x) * f'(x) + f''(x) * f''(x) = f'(x)\r<BR>\n" );
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document.write( "* Substitute x = g(3) in the above equation:<BR>\n" );
document.write( "   f'''(g(3)) * f'(g(3)) + [f''(g(3))]^2 = f'(g(3))<BR>\n" );
document.write( "   f'''(g(3)) * 2 + 1^2 = 2 <BR>\n" );
document.write( "   2 * f'''(g(3)) = 1<BR>\n" );
document.write( "   f'''(g(3)) = 1/2\r<BR>\n" );
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document.write( "* Now, substitute f'''(g(3)) = 1/2 back into the equation for g'''(3):<BR>\n" );
document.write( "   4 * (1/2) + 2 = g'''(3)<BR>\n" );
document.write( "   g'''(3) = 4\r<BR>\n" );
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document.write( "**4. Find g''''(3):**\r<BR>\n" );
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document.write( "* Differentiate the equation for g'''(x) obtained in step 3:<BR>\n" );
document.write( "   [f''''(g(x)) * g'(x)] * g'(x) + [f'''(g(x)) * g''(x)] * 2 + 2 * g'''(x) = g''''(x)\r<BR>\n" );
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document.write( "* Substitute x = 3:<BR>\n" );
document.write( "   [f''''(g(3)) * 2] * 2 + [1/2 * 2] * 2 + 2 * 4 = g''''(3)<BR>\n" );
document.write( "   4 * f''''(g(3)) + 2 + 8 = g''''(3)<BR>\n" );
document.write( "   4 * f''''(g(3)) + 10 = g''''(3)\r<BR>\n" );
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document.write( "* To find f''''(g(3)), we need to differentiate the equation f'''(x) * f'(x) + [f''(x)]^2 = f'(x) with respect to x: <BR>\n" );
document.write( "   f''''(x) * f'(x) + f'''(x) * f''(x) + 2 * f''(x) * f'''(x) = f''(x)\r<BR>\n" );
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document.write( "* Substitute x = g(3) in the above equation:<BR>\n" );
document.write( "   f''''(g(3)) * 2 + 1/2 * 1 + 2 * 1 * 1/2 = 1 <BR>\n" );
document.write( "   2 * f''''(g(3)) + 1 = 1<BR>\n" );
document.write( "   f''''(g(3)) = 0\r<BR>\n" );
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document.write( "* Substitute f''''(g(3)) = 0 back into the equation for g''''(3):<BR>\n" );
document.write( "   4 * 0 + 10 = g''''(3)<BR>\n" );
document.write( "   g''''(3) = 10\r<BR>\n" );
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document.write( "**5. Find g'''''(3):**\r<BR>\n" );
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document.write( "* Differentiate the equation for g''''(x) obtained in step 4:<BR>\n" );
document.write( "   [f'''''(g(x)) * g'(x)] * g'(x) + [f''''(g(x)) * g''(x)] * 2 + 2 * g''''(x) = g'''''(x)\r<BR>\n" );
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document.write( "* Substitute x = 3:<BR>\n" );
document.write( "   [f'''''(g(3)) * 2] * 2 + [0 * 2] * 2 + 2 * 10 = g'''''(3)<BR>\n" );
document.write( "   4 * f'''''(g(3)) + 20 = g'''''(3)\r<BR>\n" );
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document.write( "* To find f'''''(g(3)), we need to differentiate the equation f''''(x) * f'(x) + f'''(x) * f''(x) + 2 * f''(x) * f'''(x) = f''(x) with respect to x. This will involve higher-order derivatives of f(x) which are not directly provided in the given information. \r<BR>\n" );
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document.write( "**Therefore, we cannot determine the exact value of g'''''(3) with the given information.**\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "The provided information is insufficient to calculate the exact value of (d^5 g)/(dx^5) when x = 3.<BR>\n" );
document.write( " </SPAN>\n" );
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