What is the derivative of sqrt(x^2-1) / (x^2+1)√x2−1x2+1?

1 Answer
Roella W.
Jan 15, 2016

(x(-x^2 +3))/(sqrt(x^2-1)*(x^2+1)^2)x(−x2+3)√x2−1⋅(x2+1)2

Explanation:

The derivative can be found by using the quotient rule, which states that if f(x) = g(x)/(h(x))f(x)=g(x)h(x) then
f'(x) = (h(x)g'(x) - h'(x)g(x))/(h^2(x))

In this case g(x) = (x^2-1)^(1/2)
and h(x) = (x^2+1)

Using the chain rule, g'(x) = 1/2(x^2 - 1) ^(-1/2)*2x = x/sqrt(x^2-1)
h'(x) = 2x

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

=(x(x^2+1) - 2x(x^2-1))/(sqrt(x^2-1)*(x^2+1)^2)

=(x^3+x-2x^3+2x)/(sqrt(x^2-1)*(x^2+1)^2) = (x(-x^2 +3))/(sqrt(x^2-1)*(x^2+1)^2)

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