Differentiate x/(x + c/x)

Question

Differentiate 

 

x/(x + c/x)

 

 

ANSWER:  2cx / (x² + c)²

p189, #25

Answer 1

We will have to use the quotient rule, but I would "get rid of the complex fraction before starting with that.

 

     x

----------- (multiply top and bottom of the big fraction by x)

  x + c/x

 

That results in:

 

 x²/(x²+c)

 

Now use the quotient rule:

 

(x²+c)(2x) - x²(2x)

------------------------

      (x²+c)²

 

Simplify the numerator:

 

2x³+2cx - 2x³

-------------------

    (x²+c)²

 

  2cx/(x²+c)²