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$f(x)=\frac{x+2}{x^2-1}$

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Find derivatives using the quotient rule I

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Q.

f(x)=\frac{x+2}{x^2-1}

Identify Function: Let's identify the function we are differentiating. The function is $f(x) = \frac{x+2}{x^2-1}$ , which is a quotient of two functions.
Apply Quotient Rule: We will use the quotient rule to differentiate this function. The quotient rule states that if we have a function $f(x) = \frac{g(x)}{h(x)}$ , then its derivative $f'(x)$ is given by $f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2}$ .
Identify Numerator and Denominator: Let's identify $g(x)$ as the numerator, which is $x+2$ , and $h(x)$ as the denominator, which is $x^2-1$ . We will need to find the derivatives $g'(x)$ and $h'(x)$ .
Find Derivatives: The derivative of $g(x) = x+2$ is $g'(x) = 1$ , since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Apply Quotient Rule: The derivative of $h(x) = x^2-1$ is $h'(x) = 2x$ , since the derivative of $x^2$ is $2x$ and the derivative of a constant is $0$ .
Simplify Numerator: Now we can apply the quotient rule. We have $g'(x) = 1$ and $h'(x) = 2x$ , so the derivative of $f(x)$ is $f'(x) = \frac{(1 \cdot (x^2-1) - (x+2) \cdot 2x)}{(x^2-1)^2}$ .
Further Simplify Numerator: Let's simplify the numerator of the derivative. We have $f'(x) = \frac{x^2 - 1 - 2x^2 - 4x}{(x^2-1)^2}$ .
Final Derivative: Further simplifying the numerator, we get $f'(x) = \frac{-x^2 - 4x - 1}{(x^2-1)^2}$ .
Final Derivative: Further simplifying the numerator, we get $f'(x) = \frac{-x^2 - 4x - 1}{(x^2-1)^2}$ .The derivative of the function $f(x) = \frac{x+2}{x^2-1}$ is $f'(x) = \frac{-x^2 - 4x - 1}{(x^2-1)^2}$ . This is the final simplified form of the derivative.

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