Home›Calculus 1›Power rule›Problem 2.143

Derivative of \( \displaystyle \frac{5 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \)

Problem 2.143 · medium Video

Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \).

\[ \frac{d}{d x} \frac{5 \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \sqrt{\left(3 x + 2\right)^{2} + 1}}{3} \]
constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(\left(3 x + 2\right)^{2} + 1\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
chainApply the chain rule.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(3 x + 2\right)^{2}}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
constant-multipleSimplify the constant coefficient.✓ Proved
\[ = \frac{5 \left(6 x + 4\right) \frac{d}{d x} \left(3 x + 2\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
chainApply the chain rule to the inner squared term.✓ Proved
\[ = \frac{5 \left(18 x + 12\right)}{6 \sqrt{\left(3 x + 2\right)^{2} + 1}} \]
derivative algebraDifferentiate the innermost linear term. Multiply the constants in the derivative term.✓ Proved
\[ = \frac{15 x + 10}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \]
simplify rewriteSimplify the final expression by canceling the 6 and 6/3. Rewrite the negative exponent as a denominator.✓ Proved

Answer \( \frac{5 \left(3 x + 2\right)}{\sqrt{\left(3 x + 2\right)^{2} + 1}} \)

Top: the graph of f = 5sqrt((3x + 2)² + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(9x + 6)/(3sqrt((3x + 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 (3x + 2)*2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x + 2)*2 + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x + 2)*2 + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x + 2)*2 + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x + 2)*2 + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x + 2)*2 + 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x + 2)*2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where (3x + 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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-05: fail — The math is correct, and SymPy proved every line. Step 4→5 does two things: it combines 5/3·1/2 into 5/6, and it drops the +1 inside the derivative (sum/constant rule). It is labelled only 'constant-multiple', and the note mentions only the coefficient, so the sum-rule part is unnamed. It should be split into two steps or labelled more accurately.

Every verdict on record (16)

claude-sonnet-5-5: fail (style) 2026-10-05 — The math is correct, and SymPy proved every line. Step 4→5 does two things: it combines 5/3·1/2 into 5/6, and it drops the +1 inside the derivative (sum/constant rule). It is labelled only 'constant-multiple', and the note mentions only the coefficient, so the sum-rule part is unnamed. It should be split into two steps or labelled more accurately.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two changes at once: it pulls out the constant factor (constant‑multiple) and simultaneously drops the derivative of the constant term +1 (an algebraic simplification). Each step must alter only one aspect, so this step violates the contract.
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: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.
gpt-oss:20b: fail (error) 2026-09-19 — Step 5 removes the derivative of the constant term +1 while keeping the same constant factor, effectively applying two rules (constant‑multiple and derivative of a sum) in one step, which violates the one‑rule‑per‑step requirement.
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: fail 2026-09-17 — Step 5 combines two operations—simplifying the derivative of the constant term and applying the constant‑multiple rule—without separating them. Additionally, step 9’s note about “canceling the 6 and 6/3” is misleading; the simplification is simply (5/6)·6 = 5.
deepseek-r1:70b: fail 2026-09-17 — Step 9 note incorrectly states the simplification as canceling 6 and 6/3, which is misleading.

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.