Derivative of \( \displaystyle 3 \sqrt{\left(x + 1\right)^{2} + 1} \)
Problem 2.1289 · medium
Differentiate \( \displaystyle f(x) = 3 \sqrt{\left(x + 1\right)^{2} + 1} \).
- \[ \frac{d}{d x} 3 \sqrt{\left(x + 1\right)^{2} + 1} \]Start with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} \sqrt{\left(x + 1\right)^{2} + 1} \]constant rewritePull the constant out of the derivative. Rewrite the square root as a power.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(\left(x + 1\right)^{2} + 1\right)}{2 \sqrt{\left(x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{3 \left(2 x + 2\right)}{2 \sqrt{\left(x + 1\right)^{2} + 1}} \]derivative constantDifferentiate the inner function. The derivative of 1 is 0.✓ Proved
- \[ = \frac{3 \left(x + 1\right)}{\sqrt{\left(x + 1\right)^{2} + 1}} \]algebraSimplify the coefficients.✓ Proved
- \[ = \frac{3 x + 3}{\sqrt{\left(x + 1\right)^{2} + 1}} \]simplifyRewrite with a positive exponent and simplify.✓ Proved
Answer \( \frac{3 \left(x + 1\right)}{\sqrt{\left(x + 1\right)^{2} + 1}} \)
✓ 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.
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 (x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 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 (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (style) 2026-09-30 — Step 6 is labeled 'constant' but performs algebraic simplification (removing '+ 0'). The label 'constant' typically refers to the constant multiple rule or the derivative of a constant, neither of which describes simplifying an expression by dropping a zero term; 'algebra' or 'simplify' would be the correct label.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.