document.write( "Question 1088245: Find x’ for the function x(t) defined implicitly below. Compute the equation of the tangent line at the indicated point. \r
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Algebra.Com's Answer #702863 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"t%2Aln%28x%29\"\"%22%22=%22%22\"\"x%2Ae%5Et-1+\"\r\n" );
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document.write( "Differentiating implicitly to find \"%22x%27%28t%29%22\"\r\n" );
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document.write( "\"t%2A%281%2Fx%29%2A%22x%27%22%2Bln%28x%29\"\"%22%22=%22%22\"\"x%2Ae%5Et%2Be%5Et%2A%22x%27%22\"\r\n" );
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document.write( "Clear the fraction by multiplying through by LCD=x\r\n" );
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document.write( "\"t%2A%22x%27%22%2Bx%2Aln%28x%29\"\"%22%22=%22%22\"\"x%5E2%2Ae%5Et%2Bx%2Ae%5Et%2A%22x%27%22\"\r\n" );
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document.write( "\"t%2A%22x%27%22-x%2Ae%5Et%2A%22x%27%22\"\"%22%22=%22%22\"\"x%5E2%2Ae%5Et-x%2Aln%28x%29\"\r\n" );
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document.write( "\"%28t-x%2Ae%5Et%29%2A%22x%27%22\"\"%22%22=%22%22\"\"x%5E2%2Ae%5Et-x%2Aln%28x%29\"\r\n" );
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document.write( "\"%22x%27%22=%28x%5E2%2Ae%5Et-x%2Aln%28x%29%29%2F%28t-x%2Ae%5Et%29\"\r\n" );
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document.write( "When y is the dependent variable and x is the independent\r\n" );
document.write( "variable, the point-slope equation of the line through\r\n" );
document.write( "(x1,y1) with slope m is \"y-y%5B1%5D\"\"%22%22=%22%22\"\"m%28x-x%5B1%5D%29\".\r\n" );
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document.write( "But in this case, x is the dependent variable and t is the independent\r\n" );
document.write( "variable, so the equation of the line through (t1,x1)\r\n" );
document.write( "with slope m is \"x-x%5B1%5D\"\"%22%22=%22%22\"\"m%28t-t%5B1%5D%29\".\r\n" );
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document.write( "(t1,x1) = (0,1)\r\n" );
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document.write( "And the slope m is x' evaluated at this point:\r\n" );
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document.write( "\"%22x%27%280%2C1%29%22=%281%5E2%2Ae%5E0-1%2Aln%281%29%29%2F%280-1%2Ae%5E0%29\"\r\n" );
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document.write( "\"%22x%27%280%2C1%29%22\"\"%22%22=%22%22\"\"%281%2A1-1%2A0%29%2F%28-1%2A1%29\"\r\n" );
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document.write( "\"%22x%27%280%2C1%29%22\"\"%22%22=%22%22\"\"-1\"\r\n" );
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document.write( "So \"m\"\"%22%22=%22%22\"\"-1\"\r\n" );
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document.write( "So substituting m = -1, and (t1,x1) = (0,1)\r\n" );
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document.write( "\"x-x%5B1%5D\"\"%22%22=%22%22\"\"m%28t-t%5B1%5D%29\"\r\n" );
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document.write( "\"x-1\"\"%22%22=%22%22\"\"-1%28t-0%29\"\r\n" );
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document.write( "\"x-1\"\"%22%22=%22%22\"\"-t\"\r\n" );
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document.write( "\"x\"\"%22%22=%22%22\"\"-t%2B1\"\r\n" );
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document.write( "Edwin
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