document.write( "Question 1205160: Find (nth) derivative of f(x)=(ln(x))^2 \n" ); document.write( "
Algebra.Com's Answer #842049 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Sorry I'm just now posting this solution, but it just occurred to me\r\n" );
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document.write( "Let's take the first 3 derivatives.  I'm not going to\r\n" );
document.write( "show my work to get the derivatives.  The first one we use\r\n" );
document.write( "the power formula and the natural log formula. After that,\r\n" );
document.write( "we just use the natural log formula, and the quotient formula.\r\n" );
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document.write( "\"f%28x%29\"\"%22%22=%22%22\"\"%28ln%28x%29%5E%22%22%29%5E2\"\r\n" );
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document.write( "\"d%2F%28dx%29\"\"f%28x%29\"\"%22%22=%22%22\"\"%282%2Aln%28x%29%29%2Fx\"\r\n" );
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document.write( "\"d%5E2%2F%28dx%5E2%29\"\"f%28x%29\"\"%22%22=%22%22\"\"%282-2%2Aln%28x%29%29%2Fx%5E2\"\r\n" );
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document.write( "\"d%5E3%2F%28dx%5E3%29\"\"f%28x%29\"\"%22%22=%22%22\"\"%284%2Aln%28x%29-6%29%2Fx%5E3\"\r\n" );
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document.write( "Let's write those derivatives in like form:\r\n" );
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document.write( "\"d%2F%28dx%29\"\"f%28x%29\"\"%22%22=%22%22\"\"+%28-1%29%5E1%2A%28%280-2%2Aln%28x%29%29%2Fx%5E1%29\"\r\n" );
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document.write( "\"d%5E2%2F%28dx%5E2%29\"\"f%28x%29\"\"%22%22=%22%22\"\"+%28-1%29%5E2%2A%28%282-2%2Aln%28x%29%29%2Fx%5E2%29\"\r\n" );
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document.write( "\"d%5E3%2F%28dx%5E3%29\"\"f%28x%29\"\"%22%22=%22%22\"\"+%28-1%29%5E3%2A%28%286-4%2Aln%28x%29%29%2Fx%5E3%29\"\r\n" );
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document.write( "Assume the (n-1)st derivative is of the form\r\n" );
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document.write( "\"d%5E%28n-1%29%2F%28dx%5E%28n-1%29%29\"\"f%28x%29\"\"%22%22=%22%22\"\"%28-1%29%5E%28n-1%29%2A%28a%5Bn-1%5D-b%5Bn-1%5D%2Aln%28x%29%29%2Fx%5E%28n-1%29\"\r\n" );
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document.write( "Then the nth derivative is\r\n" );
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document.write( "\"d%5En%2F%28dx%5En%29\"\"f%28x%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "where the sequence {bn} is defined by \r\n" );
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document.write( "\"b%5Bn%5D\"\"%22%22=%22%22\"\"b%5Bn-1%5D%2A%28n-1%29\" where \"b%5B1%5D=2\"\r\n" );
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document.write( "and where the sequence {an} is defined by\r\n" );
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document.write( "\"a%5Bn%5D\"\"%22%22=%22%22\"\"a%5Bn-1%5D%2A%28n-1%29%2Bb%5Bn-1%5D\" where \"a%5B1%5D=0\"\r\n" );
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document.write( "Edwin
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