Derivative of \( \displaystyle \frac{5 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \)
Problem 2.141 · medium
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \).
- \[ \frac{d}{d x} \frac{5 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(4 \left(x + 1\right)^{2} + 1\right)}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{5 \frac{d}{d x} 4 \left(x + 1\right)^{2}}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} 1}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} + \frac{5 \frac{d}{d x} 4 \left(x + 1\right)^{2}}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} \]sumSplit the derivative of the sum.✓ Proved
- \[ = \frac{5 \frac{d}{d x} 4 \left(x + 1\right)^{2}}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} \]constantThe derivative of the constant 1 is 0.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(x + 1\right)^{2}}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]constant-multiple algebraPull out the constant 4. Simplify the coefficients 5/4 * 4 = 5.✓ Proved
- \[ = \frac{10 \left(x + 1\right) \frac{d}{d x} \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]powerApply the power rule to (x + 1)**2.✓ Proved
- \[ = \frac{10 \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]chainApply the chain rule to (x + 1).✓ Proved
- \[ = \frac{10 x + 10}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]simplifySimplify the expression into a single fraction.✓ Proved
Answer \( \frac{10 \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 4*(x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(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 4*(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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 5 incorrectly labels the removal of the constant term's derivative as 'sum'. The step actually applies the 'constant' rule (derivative of 1 is 0) to eliminate that term, but fails to show the intermediate split or the zero result, effectively combining the sum rule, constant rule, and simplification into one step. Step 6 then redundantly applies the sum rule to an expression that should have already been simplified in Step 5, creating a logical inconsistency in the sequence.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 incorrectly labels the removal of the constant term's derivative as 'sum'. The step actually applies the 'constant' rule (derivative of 1 is 0) to eliminate that term, but fails to show the intermediate split or the zero result, effectively combining the sum rule, constant rule, and simplification into one step. Step 6 then redundantly applies the sum rule to an expression that should have already been simplified in Step 5, creating a logical inconsistency in the sequence.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 is mathematically incorrect; it claims to apply the sum rule to remove the derivative of the constant 1, but it fails to split the expression into two terms as required by the sum rule, effectively skipping the split shown in Step 6. This violates the 'one thing per step' constraint and presents a logically invalid transition.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is mathematically incorrect; it drops the derivative of the constant term (1) from the inner function without justification, effectively skipping the application of the sum rule. Step 6 then incorrectly applies the sum rule to an expression that was already malformed in Step 5.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19 — Step 5 incorrectly labels a constant‑multiplication simplification as a sum rule; the step only simplifies 5/2·1/2 to 5/4, which should be an algebraic simplification. No mathematical errors are present.qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is incorrect because it drops the derivative of the constant term (1) from the sum without explicitly separating the terms first. The sum rule requires splitting the derivative of the sum into the sum of derivatives (as done in Step 6), but Step 5 attempts to apply the sum rule while simultaneously discarding the constant term, violating the 'one change per step' constraint and resulting in a logically invalid transition from Step 4.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 incorrectly drops the derivative of the constant term +1 in the inner function. The chain rule should produce Derivative(4*(x+1)**2 + 1, x), not just Derivative(4*(x+1)**2, x).qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 5 is mathematically incorrect. It claims to apply the sum rule to remove the derivative of the constant 1, but it fails to distribute the outer coefficient `5/4 * (4*(x + 1)**2 + 1)**(-1/2)` to the term `Derivative(1, x)`. The step effectively drops the term `5/4 * (4*(x + 1)**2 + 1)**(-1/2) * 0` without showing the distribution required by the sum rule, or incorrectly implies that `Derivative(A + B)` becomes `Derivative(A)` directly inside a product without handling the constant term's derivative explicitly in the expression structure. Specifically, line 4 is `C * Derivative(A + 1)`. Line 5 is `C * Derivative(A)`. This skips the step where `Derivative(A + 1)` becomes `Derivative(A) + Derivative(1)`. While the result is correct because the derivative of 1 is 0, the step labeled 'sum' does not perform the sum rule expansion; it performs a simplification that assumes the derivative of the constant is zero, which should be labeled 'constant' or handled in a separate step. However, looking closer at Step 5 vs Step 6: Step 5 goes from `Derivative(4*(x+1)^2 + 1)` to `Derivative(4*(x+1)^2)`. This is not the sum rule. The sum rule splits a sum into a sum of derivatives. Step 6 does the splitting. Step 5 just deletes the `+1` inside the derivative operator. This is not a valid rule application labeled 'sum'. It is an invalid step or a mislabeled simplification. Actually, Step 5 is just wrong algebraically/logically as a labeled step. You cannot go from `D(f+g)` to `D(f)` by calling it 'sum'. You must go to `D(f) + D(g)`. Step 6 does that. So Step 5 is a defect.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 5 incorrectly labels the operation as a sum rule and the note claims to apply the sum rule to the inner function, but it merely simplifies the constant factor 5/2 × 1/2. This mislabeling could mislead a student about the correct rule to use.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.