Derivative of \( \displaystyle \frac{4 x^{2} - 1}{4 x^{2} + 1} \)

Problem 2.496 · hard

1. \[ \frac{d}{d x} \frac{4 x^{2} - 1}{4 x^{2} + 1} \]
   rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(4 x^{2} - 1\right) \frac{d}{d x} \frac{1}{4 x^{2} + 1} + \frac{\frac{d}{d x} \left(4 x^{2} - 1\right)}{4 x^{2} + 1} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{\left(1 - 4 x^{2}\right) \frac{d}{d x} \left(4 x^{2} + 1\right)}{\left(4 x^{2} + 1\right)^{2}} + \frac{\frac{d}{d x} \left(4 x^{2} - 1\right)}{4 x^{2} + 1} \]
   chainApply the chain rule to the second term.✓ Proved
4. \[ = \frac{8 x \left(1 - 4 x^{2}\right)}{\left(4 x^{2} + 1\right)^{2}} + \frac{8 x}{4 x^{2} + 1} \]
   derivativeDifferentiate the polynomial terms.✓ Proved
5. \[ = - \frac{8 x \left(4 x^{2} - 1\right)}{\left(4 x^{2} + 1\right)^{2}} + \frac{8 x}{4 x^{2} + 1} \]
   algebraSimplify the expression using negative exponents.✓ Proved
6. \[ = \frac{- 8 x \left(4 x^{2} - 1\right) + 8 x \left(4 x^{2} + 1\right)}{\left(4 x^{2} + 1\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
7. \[ = \frac{16 x}{\left(4 x^{2} + 1\right)^{2}} \]
   algebra simplifyDistribute the 8*x in the numerator. Simplify the numerator by combining like terms.✓ Proved