Find Derivative of f(x) = x/(x+c/x): Step-by-Step

[nothing123]

find the derivative of the following:
f(x) = x/(x+c/x)

through simplication, i got:
x(x+cx^-1)^-1
=x(x^-1 + c^-1*x)
=1+1/c*x^2

taking the derivative,
-c^-2*x^2 + c^-1*2x

its wrong though so where did i go wrong?

 

[neutrino]

through simplication, i got:
x(x+cx^-1)^-1
=x(x^-1 + c^-1*x)
=1+1/c*x^2

That's not right. (x+y)^-1 is not the same as x^-1 + y^-1.

Without simplifying the original expression you could have used the quotient rule to find the derivative, OR you could differentiate x(x+cx^-1)^-1 using the product rule.

 

[HallsofIvy]

I would be inclined to multiply both numerator and denominator of the function by x:
$$f(x)= \frac{x}{x+ \frac{c}{x}}= \frac{x^2}{x^2+ c}$$
for all x except 0.