Derivative of \( \displaystyle - \frac{\sqrt{9 x^{2} + 1}}{3} \)

Problem 2.1185 · medium

1. \[ \frac{d}{d x} \left(- \frac{\sqrt{9 x^{2} + 1}}{3}\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \sqrt{9 x^{2} + 1}}{3} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \left(9 x^{2} + 1\right)}{6 \sqrt{9 x^{2} + 1}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} 9 x^{2}}{6 \sqrt{9 x^{2} + 1}} \]
   algebraDistribute the derivative over the sum.✓ Proved
5. \[ = - \frac{\frac{d}{d x} 9 x^{2}}{6 \sqrt{9 x^{2} + 1}} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = - \frac{3 x}{\sqrt{9 x^{2} + 1}} \]
   power algebra simplifyDifferentiate the power term. Combine the terms into a single fraction. Simplify the final expression.✓ Proved