Step 1: Identify the function and the differentiation rule needed. The given function is f(x) = (x² − 1)/(x² + 1). This is a quotient of two functions, so we will use the quotient rule: If f(x) = (g(x))/(h(x)), then f'(x) = (g'(x)h(x) − g(x)h'(x))/((h(x))²). Step 2: Define g(x) and h(x) and find their derivatives. Let g(x) = x² − 1. Then g'(x) = d/dx(x² − 1) = 2x. Let h(x) = x² + 1. Then h'(x) = d/dx(x² + 1) = 2x. Step 3: Apply the quotient rule. Substitute g(x), h(x), g'(x), and h'(x) into the quotient rule formula: f'(x) = ((2x)(x² + 1) − (x² − 1)(2x))/((x² + 1)²) Step 4: Simplify the numerator. f'(x) = (2x(x² + 1 − (x² − 1)))/((x² + 1)²); f'(x) = (2x(x² + 1 − x² + 1))/((x² + 1)²); f'(x) = (2x(2))/((x² + 1)²); f'(x) = (4x)/((x² + 1)²) Final answer: (4x)/((x² + 1)²)

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March 22, 2026Mathematics

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Answer

\frac{4x}{(x^2 + 1)^2}

Step-by-step solution

4 steps

Identify the function and the differentiation rule needed.
The given function is $f(x) = \frac{x^2 - 1}{x^2 + 1}$ .
This is a quotient of two functions, so we will use the quotient rule:
If $f(x) = \frac{g(x)}{h(x)}$ , then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$ .
Define $g(x)$ and $h(x)$ and find their derivatives.
Let $g(x) = x^2 - 1$ .
Then $g'(x) = \frac{d}{dx}(x^2 - 1) = 2x$ .

Let $h(x) = x^2 + 1$ .
Then $h'(x) = \frac{d}{dx}(x^2 + 1) = 2x$ .
Apply the quotient rule.
Substitute $g(x)$ , $h(x)$ , $g'(x)$ , and $h'(x)$ into the quotient rule formula:
$f'(x) = \frac{(2x)(x^2 + 1) - (x^2 - 1)(2x)}{(x^2 + 1)^2}$
Simplify the numerator.
$f'(x) = \frac{2x(x^2 + 1 - (x^2 - 1))}{(x^2 + 1)^2}$ $f'(x) = \frac{2x(x^2 + 1 - x^2 + 1)}{(x^2 + 1)^2}$ $f'(x) = \frac{2x(2)}{(x^2 + 1)^2}$ $f'(x) = \frac{4x}{(x^2 + 1)^2}$
Final answer: $\boxed{\frac{4x}{(x^2 + 1)^2}}$

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Answer

Step-by-step solution

Identify the function and the differentiation rule needed.

Define g(x)g(x)g(x) and h(x)h(x)h(x) and find their derivatives.

Apply the quotient rule.

Simplify the numerator.

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Define $g(x)$ and $h(x)$ and find their derivatives.