Differentiate the following function?
g(x)=x^2=1/x^2-1
To differentiate the function g(x) = x^2 + 1 / (x^2 - 1), we can use the quotient rule. The quotient rule states that for a function h(x) = f(x) / g(x), the derivative is given by:
h'(x) = (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2
Let's apply this rule to differentiate g(x).
First, we need to find the derivatives of the numerator and denominator separately.
f(x) = x^2 + 1
g(x) = x^2 - 1
Now, let's find the derivatives of f(x) and g(x).
f'(x) = 2x (Derivative of x^2 is 2x)
g'(x) = 2x (Derivative of x^2 is 2x)
Now, we can substitute these derivatives into the quotient rule formula:
g'(x) = [(2x * (x^2 - 1)) - ((x^2 + 1) * 2x)] / (x^2 - 1)^2
Simplifying the numerator:
g'(x) = [2x^3 - 2x - 2x^3 - 2x] / (x^2 - 1)^2
= [-4x] / (x^2 - 1)^2
Therefore, the derivative of g(x) is:
g'(x) = -4x / (x^2 - 1)^2