Derivative of \( \displaystyle \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \)
Problem 2.712 · medium Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \).
- \[ \frac{d}{d x} \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \]constant-multiple rewriteMove the constant factor outside. Rewrite the square root as a power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(4 x - 3\right)^{2} + 1\right)}{4 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(4 x - 3\right)^{2}}{4 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]sum rewriteDifferentiate the sum inside the parenthesis. Rewrite the power using the exponential-logarithm identity.✓ Proved
- \[ = \frac{\left(4 x - 3\right)^{2} \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{2 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]exponential rewriteDifferentiate the exponential function. Substitute the exponential back to its power form.✓ Proved
- \[ = \frac{\left(4 x - 3\right) \frac{d}{d x} \left(4 x - 3\right)}{2 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]logarithmicDifferentiate the natural logarithm.✓ Proved
- \[ = \frac{2 \left(4 x - 3\right)}{\sqrt{\left(4 x - 3\right)^{2} + 1}} \]algebraSimplify the algebraic terms.✓ Proved
- \[ = \frac{8 x - 6}{\sqrt{\left(4 x - 3\right)^{2} + 1}} \]simplifyFinal simplified form.✓ Proved
Answer \( \frac{2 \left(4 x - 3\right)}{\sqrt{\left(4 x - 3\right)^{2} + 1}} \)
✓ 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. 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 (4*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where (4*x - 3)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where (4*x - 3)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**2 + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 3)**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 - 3)**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 5 applies the sum rule but drops the derivative of the constant term +1, so the expression is incorrect. The derivative of (4*x-3)**2 + 1 should include 0 from the constant, which is missing. This makes the subsequent steps based on an incorrect intermediate result.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] Step 5 applies the sum rule to differentiate the inner function but fails to account for the derivative of the constant term 1, which is 0. While the result is numerically correct because the derivative of the constant is zero, the step label 'sum' implies applying the rule to the sum, yet the constant term vanishes without explicit differentiation or justification in the transition. More critically, Step 6 rewrites (4*x - 3)**2 as exp(2 * log(4*x - 3)), which is invalid for negative values of (4*x - 3) without complex numbers, and the subsequent differentiation in Step 7 and 9 relies on this invalid rewrite for the general domain. However, the most glaring defect is in Step 5: the transition from Derivative((4*x - 3)**2 + 1, x) to Derivative((4*x - 3)**2, x) is an algebraic simplification (dropping a term with zero derivative), not the application of the 'sum' rule which would produce Derivative((4*x - 3)**2, x) + Derivative(1, x). The label 'sum' is incorrect for a step that simply drops a constant term inside a derivative operator without showing the sum expansion. Furthermore, the rewrite in step 6 is mathematically suspect for real-valued calculus without absolute values, but the primary defect is the mislabeling of the constant removal in step 5 as 'sum' when it's effectively an algebraic simplification or constant rule application that isn't shown.
Senior review claude-sonnet-5-5, 2026-10-12: pass — Every line is correct and was proved equal to the previous one, and the final result matches the stated answer. The exp/log detour for differentiating (4x-3)^2 is roundabout, but it is valid and every rule name is acceptable.
gpt-oss:20b: dismiss — The derivative of the constant 1 is 0, so dropping it is the sum rule applied with a vanishing term; the line is mathematically correct and SymPy proved the equality.qwen3.6:27b-mlx: dismiss — The 'sum' label is legitimate because the sum rule is applied and the constant term's zero derivative is part of the same step. The exp(2*log(...)) domain concern is a recorded domain restriction, not a defect, and the identity exp(2*log(u)) = u^2 holds wherever u is nonzero.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-12 — Every line is correct and was proved equal to the previous one, and the final result matches the stated answer. The exp/log detour for differentiating (4x-3)^2 is roundabout, but it is valid and every rule name is acceptable.qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] Step 5 applies the sum rule to differentiate the inner function but fails to account for the derivative of the constant term 1, which is 0. While the result is numerically correct because the derivative of the constant is zero, the step label 'sum' implies applying the rule to the sum, yet the constant term vanishes without explicit differentiation or justification in the transition. More critically, Step 6 rewrites (4*x - 3)**2 as exp(2 * log(4*x - 3)), which is invalid for negative values of (4*x - 3) without complex numbers, and the subsequent differentiation in Step 7 and 9 relies on this invalid rewrite for the general domain. However, the most glaring defect is in Step 5: the transition from Derivative((4*x - 3)**2 + 1, x) to Derivative((4*x - 3)**2, x) is an algebraic simplification (dropping a term with zero derivative), not the application of the 'sum' rule which would produce Derivative((4*x - 3)**2, x) + Derivative(1, x). The label 'sum' is incorrect for a step that simply drops a constant term inside a derivative operator without showing the sum expansion. Furthermore, the rewrite in step 6 is mathematically suspect for real-valued calculus without absolute values, but the primary defect is the mislabeling of the constant removal in step 5 as 'sum' when it's effectively an algebraic simplification or constant rule application that isn't shown.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies the sum rule but drops the derivative of the constant term +1, so the expression is incorrect. The derivative of (4*x-3)**2 + 1 should include 0 from the constant, which is missing. This makes the subsequent steps based on an incorrect intermediate result.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.