document.write( "Question 990668: Differentiate with respect to x
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Algebra.Com's Answer #610667 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"y\"\"%22%22=%22%22\"\"x%5E2%2Fsqrt%28x%5E2-1%29\"\r\n" );
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document.write( "\"y\"\"%22%22=%22%22\"\"x%5E2%2F%28x%5E2-1%29%5E%281%2F2%29\"\r\n" );
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document.write( "We could differentiate it as a quotient but it's a \r\n" );
document.write( "little easier to first turn it into a product:\r\n" );
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document.write( "\"y\"\"%22%22=%22%22\"\"x%5E2%28x%5E2-1%29%5E%28-1%2F2%29\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%5E2%28-1%2F2%29%28x%5E2-1%29%5E%28-3%2F2%29%282x%29\"\"%22%22%2B%22%22\"\"%28x%5E2-1%29%5E%28-1%2F2%29%2A%282x%29+\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%5E2%28-1%2Fcross%282%29%29%28x%5E2-1%29%5E%28-3%2F2%29%28cross%282%29x%29\"\"%22%22%2B%22%22\"\"%28x%5E2-1%29%5E%28-1%2F2%29%2A%282x%29+\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"-x%5E3%28x%5E2-1%29%5E%28-3%2F2%29\"\"%22%22%2B%22%22\"\"%28x%5E2-1%29%5E%28-1%2F2%29%2A%282x%29+\"\r\n" );
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document.write( "A clever trick we often play here is to factor out the smallest exponent, the\r\n" );
document.write( "\"-3%2F2\" power.\r\n" );
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document.write( "When we factor the \"-3%2F2\" power out of the \"-1%2F2\" power what's left inside is \r\n" );
document.write( "the difference of the exponents \"%28-1%2F2%29-%28-3%2F2%29=-1%2F2%2B3%2F2=2%2F2=1\"\r\n" );
document.write( "It's just the first power!\r\n" );
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document.write( "We factor \"x%28x%5E2-1%29%5E%28-3%2F2%29\" out:\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%28x%5E2-1%29%5E%28-3%2F2%29+%28-x%5E2%2B2%28x%5E2-1%29%5E1%29+\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%28x%5E2-1%29%5E%28-3%2F2%29+%28-x%5E2%2B2%28x%5E2-1%29%5E%22%22%29+\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%28x%5E2-1%29%5E%28-3%2F2%29+%28-x%5E2%2B2x%5E2-2%29+\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%28x%5E2-1%29%5E%28-3%2F2%29+%28x%5E2-2%29+\"\r\n" );
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document.write( "We write the negative exponent in the bottom as a positive exponent:\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29\"\"%22%22=%22%22\"\"x%28x%5E2-2%29%2F%28x%5E2-1%29%5E%283%2F2%29+\"\r\n" );
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document.write( "Edwin
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