Derivative of \( \displaystyle \frac{x^{2} - 1}{x^{2} + 1} \)

Problem 2.346 · hard

1. \[ \frac{d}{d x} \frac{x^{2} - 1}{x^{2} + 1} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(x^{2} - 1\right) \frac{d}{d x} \frac{1}{x^{2} + 1} + \frac{\frac{d}{d x} \left(x^{2} - 1\right)}{x^{2} + 1} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{2 x}{x^{2} + 1} + \left(x^{2} - 1\right) \frac{d}{d x} \frac{1}{x^{2} + 1} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \frac{2 x}{x^{2} + 1} + \frac{\left(1 - x^{2}\right) \frac{d}{d x} \left(x^{2} + 1\right)}{\left(x^{2} + 1\right)^{2}} \]
   chainApply the chain rule to the second part.✓ Proved
5. \[ = \frac{2 x \left(1 - x^{2}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{x^{2} + 1} \]
   derivativeDifferentiate the inner function of the chain rule.✓ Proved
6. \[ = - \frac{2 x \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} + \frac{2 x}{x^{2} + 1} \]
   algebraSimplify the expression by distributing the negative sign and exponents.✓ Proved
7. \[ = \frac{- 2 x \left(x^{2} - 1\right) + 2 x \left(x^{2} + 1\right)}{\left(x^{2} + 1\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
8. \[ = \frac{4 x}{\left(x^{2} + 1\right)^{2}} \]
   algebra simplifyExpand the numerator. Combine like terms to get the final answer.✓ Proved