Derivatives Find and simplify the derivative of the following functions.
g(x) = e^x / x²-1

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The function given is \( g(x) = \frac{e^x}{x^2 - 1} \). This is a quotient of two functions, so we will use the Quotient Rule.
2. Step 2: Recall the Quotient Rule. The Quotient Rule states that if you have a function \( h(x) = \frac{f(x)}{g(x)} \), then its derivative \( h'(x) \) is given by \( h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).
3. Step 3: Identify \( f(x) \) and \( g(x) \) in the function \( g(x) = \frac{e^x}{x^2 - 1} \). Here, \( f(x) = e^x \) and \( g(x) = x^2 - 1 \).
4. Step 4: Differentiate \( f(x) \) and \( g(x) \). The derivative of \( f(x) = e^x \) is \( f'(x) = e^x \). The derivative of \( g(x) = x^2 - 1 \) is \( g'(x) = 2x \).
5. Step 5: Apply the Quotient Rule. Substitute \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) into the Quotient Rule formula: \( g'(x) = \frac{e^x(x^2 - 1) - e^x(2x)}{(x^2 - 1)^2} \). Simplify the expression to find the derivative.