Derivative of \( \displaystyle \frac{\sqrt{9 x^{2} + 12 x + 5}}{3} \)
Problem 2.2263 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt{9 x^{2} + 12 x + 5}}{3} \).
- \[ \frac{d}{d x} \frac{\sqrt{9 x^{2} + 12 x + 5}}{3} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sqrt{9 x^{2} + 12 x + 5}}{3} \]constant-multiple rewritePull out the constant factor 1/3. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(9 x^{2} + 12 x + 5\right)}{6 \sqrt{9 x^{2} + 12 x + 5}} \]chain algebraApply the chain rule. Multiply the constants.✓ Proved
- \[ = \frac{\frac{d}{d x} 5 + \frac{d}{d x} 12 x + \frac{d}{d x} 9 x^{2}}{6 \sqrt{9 x^{2} + 12 x + 5}} \]sumDifferentiate the sum term by term.✓ Proved
- \[ = \frac{18 x + 12}{6 \sqrt{9 x^{2} + 12 x + 5}} \]derivative algebra algebraCompute the derivatives of each term. Simplify the derivative of the polynomial. Combine the terms into a single fraction.✓ Proved
- \[ = \frac{3 x + 2}{\sqrt{9 x^{2} + 12 x + 5}} \]simplifySimplify the fraction by dividing the numerator and denominator by 6.✓ Proved
Answer \( \frac{3 x + 2}{\sqrt{9 x^{2} + 12 x + 5}} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 9*x**2 + 12*x + 5 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the constant multiple rule, chain rule, and sum rule in separate steps. All labels are valid and the algebraic simplifications are correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-17 with SymPy 1.14.0.