Mindrust Posted May 20, 2011 Share Posted May 20, 2011 (edited) Find the nth derivative of sqr(2x-1). That means we need to find a pattern. Here's what I've got so far, from 1st derivative to 5th: The pattern for the exponent is simple, 1/2 - n The pattern for the co-efficient is what I'm stuck on. Here's what it looks like. I'm not sure how to express it mathematically. 1 -> 1 = 1 2 -> -1 = 1(-1) 3 -> 3 = 1(-1)(-3) 4 -> -15 = 1(-1)(-3)(-5) 5 -> 105 = 1(-1)(-3)(-5)(-7) and so on... Anyone know how to make a function for the coefficient in terms of n? Edited May 20, 2011 by Mindrust Link to comment Share on other sites More sharing options...
Bignose Posted May 20, 2011 Share Posted May 20, 2011 [math] \prod_{i=1}^n (-1)^{(i-1)}(2i-1) [/math] Link to comment Share on other sites More sharing options...
DrRocket Posted May 20, 2011 Share Posted May 20, 2011 Find the nth derivative of sqr(2x-1). That means we need to find a pattern. Here's what I've got so far, from 1st derivative to 5th: The pattern for the exponent is simple, 1/2 - n The pattern for the co-efficient is what I'm stuck on. Here's what it looks like. I'm not sure how to express it mathematically. 1 -> 1 = 1 2 -> -1 = 1(-1) 3 -> 3 = 1(-1)(-3) 4 -> -15 = 1(-1)(-3)(-5) 5 -> 105 = 1(-1)(-3)(-5)(-7) and so on... Anyone know how to make a function for the coefficient in terms of n? [math] \frac{d}{dx} (2x-1)^{\frac{1}{2}} = (2x-1)^{\frac {-1}{2}}[/math] [math] \frac{d^2}{dx^2} (2x-1)^{\frac{1}{2}} = -(2x-1)^{\frac {-3}{2}}[/math] [math]\frac{d^3}{dx^3} (2x-1)^{\frac{1}{2}} = 3(2x-1)^{\frac {-5}{2}}[/math] . . . [math]\frac{d^n}{dx^n} (2x-1)^{\frac{1}{2}} = ( (-1)^{n}\displaystyle\prod_{k=0}^{n-1} (2k-1) )(2x-1)^{\frac {-2n+1}{2}} [/math] Link to comment Share on other sites More sharing options...
Mindrust Posted May 20, 2011 Author Share Posted May 20, 2011 (edited) Thanks. BTW, what does that big "pi" looking symbol mean? Edited May 20, 2011 by Mindrust Link to comment Share on other sites More sharing options...
DJBruce Posted May 20, 2011 Share Posted May 20, 2011 (edited) Thanks. BTW, what does that big "pi" looking symbol mean? It is similar to summation notation, but instead of adding each terms you are going to multiply them together. So for example, [math]\prod_{k=1}^n \frac{1}{x_{k}}=1*\frac{1}{2}*\frac{1}{3}*...*\frac{1}{n} [/math] Edited May 20, 2011 by DJBruce Link to comment Share on other sites More sharing options...
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