document.write( "Question 71523: find the derivative of 2(pi)(r)-(2v/r^2) \n" ); document.write( "

Algebra.Com's Answer #51153 by bucky(2189) $\"\"$ $\"About$

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$\"2%28pi%29%28r%29-%282v%2Fr%5E2%29\"$
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\n" ); document.write( "The problem asks for you to take the derivative of the above expression. I presume that
\n" ); document.write( "the derivative is with respect to r and that v is a constant.
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\n" ); document.write( "One term at a time. The product $\"2%28pi%29\"$ is a constant so it is a multiplier of the derivative
\n" ); document.write( "of r. The rule that applies is that the derivative of $\"r%5En\"$ is $\"n%2Ar%5E%28n-1%29\"$
\n" ); document.write( "In this first term the exponent of r is 1. So you are taking the derivative of $\"r%5E1\"$ which
\n" ); document.write( "the rule tells you in $\"1%2Ar%5E%281-1%29\"$ and this simplifies to $\"1%2Ar%5E0+=+1%2A1+=+1\"$ .
\n" ); document.write( "Putting this all together for the first term, the derivative is $\"2%2A%28pi%29%2A1+=+2%2A%28pi%29\"$ .
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\n" ); document.write( "On to the second term. The most critical point here is to recognize that a positive
\n" ); document.write( "exponent in the denominator is equivalent to a negative exponent in the numerator.
\n" ); document.write( "Using this we can convert $\"%282v%29%2Fr%5E2\"$ to an equivalent form $\"%282v%29%2Ar%5E%28-2%29\"$ .
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\n" ); document.write( "Now we can apply the same technique as we did for the first term. The factor (2v) is
\n" ); document.write( "presumed to be a constant and therefore will be a multiplier of the derivative of $\"r%5E%28-2%29\"$ .
\n" ); document.write( "Again we use the rule that the derivative of $\"r%5En\"$ is $\"n%2Ar%5E%28n-1%29\"$ .
\n" ); document.write( "So the
\n" ); document.write( "derivative of $\"r%5E%28-2%29\"$ is $\"%28-2%29%2Ar%5E%28-2-1%29\"$ which simplifies to $\"%28-2%29%2Ar%5E%28-3%29\"$ .
\n" ); document.write( "Don't forget that this gets multiplied by the constant (2v) so that the derivative
\n" ); document.write( "for this second term is $\"%282v%29%28-2%29r%5E%28-3%29\"$ . Multiplying this out results in:
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\n" ); document.write( " $\"%28-4vr%5E%28-3%29%29\"$ and since a negative exponent in the numerator becomes a positive
\n" ); document.write( "exponent in the denominator, we could also write the derivative as:
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\n" ); document.write( " $\"-4v%2F%28r%5E3%29\"$ .
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\n" ); document.write( "Finally we combine the derivatives for the first and second terms to get:
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\n" ); document.write( " $\"2%28pi%29+-+%28-4v%29%2Fr%5E3\"$
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\n" ); document.write( "and taking care of the signs for the second term concludes the effort by producing
\n" ); document.write( "the result:
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\n" ); document.write( " $\"2%28pi%29+%2B+%284v%29%2Fr%5E3\"$
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\n" ); document.write( "Hope this helps you to see your way through the problem. \n" ); document.write( "

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