My Math Forum How do you find V prime of r for this function?

 Calculus Calculus Math Forum

October 1st, 2017, 03:24 PM   #1
Newbie

Joined: Oct 2017
From: ...

Posts: 5
Thanks: 0

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 Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top