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October 1st, 2017, 03:24 PM   #1
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How do you find V prime of r for this function?

I'm really confused about how I am meant to solve this question. I'm assuming that I need to use the quotient rule but because everything involved is a variable I'm really confused.
Attached Images functions.jpg (21.1 KB, 10 views) October 1st, 2017, 04:15 PM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Every thing is not a variable! (or, more grammatically, "not everything is a variable") "r" is the only variable. The problem specifically says that $\displaystyle \epsilon$ and R are constants. The function is $\displaystyle V(r)= \epsilon\left(\left(\frac{R}{r}\right)^{12}- 2\left(\frac{R}{r}\right)^6\right)$ If you let $\displaystyle x= \frac{R}{r}$ then $\displaystyle V= \epsilon\left(x^{12}- x^6\right)$ so that $\displaystyle \frac{dV}{dx}= \epsilon\left(12x^{11}- 12x^5\right)$. By the chain rule, $\displaystyle \frac{dV}{dr}= \frac{dV}{dx}\frac{dx}{dr}$. The derivative of $\displaystyle x= \frac{R}{r}$, with respect to r, using the quotient rule, is $\displaystyle \frac{dx}{dr}\frac{0(r)- R(1)}{r^2}= -\frac{R}{r^2}$. So $\displaystyle \frac{dV}{dr}= V'= \epsilon\left(12\left(\frac{R}{r}\right)^{11}- 12\left(\frac{R}{r}\right)^5\right)\left(\frac{-R}{r^2}\right)$. By the way- you might find it easier to deal with fractions by writing the denominator in the numerator with a negative power. For example, here, $\displaystyle V(r)= \epsilon\left(R^{12}r^{-12}- 2R^6r^{-6}\right)$ and then, just using the power formula, $\displaystyle V'= \epsilon\left(-12R^{12}r^{-13}+ 12R^6r^{-7}\right)$. With a little algebra, you can show that this is the same answer as before. Last edited by Country Boy; October 1st, 2017 at 04:44 PM. October 1st, 2017, 04:36 PM #3 Newbie   Joined: Oct 2017 From: ... Posts: 5 Thanks: 0 Wow, thanks so much, it makes a lot more sense now. I should probably practice more though. Thanks again! Tags find, function, prime Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post burgess Algebra 9 August 28th, 2014 08:04 AM harrypham Number Theory 36 February 11th, 2013 01:46 AM Bogauss Number Theory 22 October 17th, 2012 01:27 PM kriegor Number Theory 18 April 23rd, 2012 11:25 AM Jamers328 Number Theory 10 October 11th, 2007 04:04 AM

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