Find the derivative of the function

Question

So the function given was (x²+1)/x and I was really unsure how to approach it. I tried to split the fraction but that didn't work, so then I tried to take the derivative of the numerator and I got 2x and then I just divided by the denominator and was left with 2, but this seemed wrong. Could you please help me find this derivative. Help is much appreciated.

Answer 1

Hi Phil,

 

First you need to understand that something like f(x) = 1/x² can be written as f(x) = x^-2. In that case f'(x) = -2x^-3 (or -2/x³).

 

For the problem in question your idea to "split" the function would have worked. f(x) = x + 1/x or x + x^-1. Now f'(x) = 1-x^-2 (1-1/x²).

 

You could also have employed the quotient rule (denominator times derivative of the numerator minus numerator times derivative of the denominator all over the denominator squared).

 

f'(x) = [x(2x) - (x²+1)(1)]/x²

 

       = (2x² -x² - 1)/x²

 

       = (x²-1)/x² 

 

       Or if you want 1 - 1/x²

 

Same result.