SOLUTION: find the derivative of 2(pi)(r)-(2v/r^2)

Question 71523: find the derivative of 2(pi)(r)-(2v/r^2)
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!

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The problem asks for you to take the derivative of the above expression. I presume that
the derivative is with respect to r and that v is a constant.
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One term at a time. The product is a constant so it is a multiplier of the derivative
of r. The rule that applies is that the derivative of is
In this first term the exponent of r is 1. So you are taking the derivative of which
the rule tells you in and this simplifies to .
Putting this all together for the first term, the derivative is .
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On to the second term. The most critical point here is to recognize that a positive
exponent in the denominator is equivalent to a negative exponent in the numerator.
Using this we can convert to an equivalent form .
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Now we can apply the same technique as we did for the first term. The factor (2v) is
presumed to be a constant and therefore will be a multiplier of the derivative of .
Again we use the rule that the derivative of is .
So the
derivative of is which simplifies to .
Don't forget that this gets multiplied by the constant (2v) so that the derivative
for this second term is . Multiplying this out results in:
.
and since a negative exponent in the numerator becomes a positive
exponent in the denominator, we could also write the derivative as:
.
.
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Finally we combine the derivatives for the first and second terms to get:
.

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and taking care of the signs for the second term concludes the effort by producing
the result:
.

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Hope this helps you to see your way through the problem.
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