Home›Calculus 1›Power rule›Problem 2.155

Derivative of \( \displaystyle 3 \sqrt{\left(x + 2\right)^{2} + 1} \)

Problem 2.155 · medium Video

Differentiate \( \displaystyle f(x) = 3 \sqrt{\left(x + 2\right)^{2} + 1} \).

\[ \frac{d}{d x} 3 \sqrt{\left(x + 2\right)^{2} + 1} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 3 \frac{d}{d x} \sqrt{\left(x + 2\right)^{2} + 1} \]
constant-multiple rewritePull the constant factor out of the derivative. Rewrite the square root as a fractional power.✓ Proved
\[ = \frac{3 \frac{d}{d x} \left(\left(x + 2\right)^{2} + 1\right)}{2 \sqrt{\left(x + 2\right)^{2} + 1}} \]
chainApply the chain rule.✓ Proved
\[ = \frac{3 \left(2 x + 4\right)}{2 \sqrt{\left(x + 2\right)^{2} + 1}} \]
derivative simplifyDifferentiate the inner function. Simplify the derivative of the inner term.✓ Proved
\[ = \frac{3 x + 6}{\sqrt{\left(x + 2\right)^{2} + 1}} \]
algebra rewriteCancel the 2 and the 1/2. Rewrite the negative exponent as a fraction.✓ Proved

Answer \( \frac{3 \left(x + 2\right)}{\sqrt{\left(x + 2\right)^{2} + 1}} \)

Top: the graph of f = 3sqrt((x + 2)² + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3(x + 2)/sqrt((x + 2)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. 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 (x + 2)**2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where (x + 2)**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: fail (error) — Step 4 applies both the power rule and the chain rule in one go, violating the rule‑granularity requirement. Each step must change only one thing using a single named rule.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-05: pass — All steps are correct and SymPy proved each one. The rule labels are legitimate and each step changes one thing. Step 5 applies the derivative rule to the inner function in a single change, and step 6 only removes the '*1' and '+0' terms.

gpt-oss:20b: dismiss — Differentiating the outer power u^(1/2) to (1/2)u^(-1/2) times u' is the chain rule applied to a power function, which is one step; the label 'chain' is legitimate and the contract allows evaluating the outer derivative as part of it.

Every verdict on record (16)

claude-sonnet-5-5: pass 2026-10-05 — All steps are correct and SymPy proved each one. The rule labels are legitimate and each step changes one thing. Step 5 applies the derivative rule to the inner function in a single change, and step 6 only removes the '*1' and '+0' terms.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies both the power rule and the chain rule in one go, violating the rule‑granularity requirement. Each step must change only one thing using a single named rule.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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 2026-09-26 with SymPy 1.14.0.