Question

How do you differentiate `f(x) = x^2 sqrt((x+1)/(x^2+1))`?

Answer 1

`f'(x)=(x^2)(((x^2+1)^(1/2)(1/(2sqrt(x+1)))-(x+1)^(1/2)((x)/(sqrt(x^2+1))))/(x^2+1)) + (2x)(sqrt((x+1)/(x^2+1)))`

Explanation:

For this problem, we will need a combination of chain rule, product rule, quotient rule, and power rule...Yuck.

`f'(x)=(x^2)(sqrt((x+1)/(x^2+1)))' + (2x)(sqrt((x+1)/(x^2+1)))`

Notice the little apostrophe tagging along the first `sqrt((x+1)/(x^2+1)`? That means we still need to find the derivative of that part. Start off with chain rule...

`1/(2sqrt((x+1)/(x^2+1))) * (((x+1)^(1/2))/(x^2+1)^(1/2))`

The first part of the above looks good. Now let's focus on the last part. Using quotient rule...

`((x^2+1)^(1/2)(1/(2sqrt(x+1)))-(x+1)^(1/2)((x)/(sqrt(x^2+1))))/(x^2+1)`

Dang...I would like to see anyone simplify this...your answer , by the way, is

`f'(x)=(x^2)(((x^2+1)^(1/2)(1/(2sqrt(x+1)))-(x+1)^(1/2)((x)/(sqrt(x^2+1))))/(x^2+1)) + (2x)(sqrt((x+1)/(x^2+1)))`

Many apologies that I can't go any further into this problem. This is certainly a mammoth...just hope that something like this doesn't appear on your tests. :)