Derivative of \( \displaystyle \ln{\left(2 \sqrt{x^{2} + 1} \right)} \)
Problem 2.724 · hard Beautiful Video
Differentiate \( \displaystyle f(x) = \ln{\left(2 \sqrt{x^{2} + 1} \right)} \).
- \[ \frac{d}{d x} \ln{\left(2 \sqrt{x^{2} + 1} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(\ln{\left(\sqrt{x^{2} + 1} \right)} + \ln{\left(2 \right)}\right) \]algebra algebraUse the logarithm product rule to split the terms. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(2 \right)} + \frac{d}{d x} \ln{\left(\sqrt{x^{2} + 1} \right)} \]sumApply the sum rule for derivatives.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\sqrt{x^{2} + 1} \right)} \]constant rewriteThe derivative of the constant log(2) is zero. Rewrite the power using the exponential and logarithm identity.✓ Proved
- \[ = \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]chainApply the chain rule to the composition of functions.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} x^{2}}{2 \left(x^{2} + 1\right)} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = \frac{x}{x^{2} + 1} \]derivative constant simplifyDifferentiate the terms inside the parentheses. The derivative of the constant 1 is zero. Simplify the final expression by canceling the 2.✓ Proved
Answer \( \frac{x}{x^{2} + 1} \)
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. A reviewer's objection to the explanation stands; see the receipt. 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 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**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 + 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 (misleading) — Step 7 incorrectly labels the simplification of log(exp(u)) as a chain rule application. The derivative of log(exp(u)) is simply u', so the step should be a rewrite, not a chain rule. This mislabeling could mislead a student into thinking a chain rule is needed here.qwen3.6:27b-mlx: fail (error) — Step 6 applies the identity log(a^b) = b*log(a) but labels it 'rewrite' while Step 7 labels the resulting simplification 'chain'. More critically, Step 7 claims to apply the chain rule to log(exp(...)), but the transition from Step 6 to Step 7 is actually a simplification using log(exp(u)) = u, not a differentiation step. The label 'chain' is incorrect for an algebraic simplification step. Furthermore, Step 6 itself is an algebraic rewrite, but the note in Step 7 is misleading because the step from 6 to 7 is not a differentiation application of the chain rule, but rather a simplification of the logarithm of an exponential. The derivative is not taken in step 7; the expression is simplified. The derivative is taken in step 8. Thus, step 7's label 'chain' is wrong because no differentiation occurs in that step; it is an algebraic simplification.
Senior review claude-sonnet-5-5, 2026-10-12: fail — Every line is mathematically correct, but line 7 is labelled and described as a chain-rule step when it is just log(exp(u)) = u, which would teach students that the chain rule is what removes log(exp()). The route is also needlessly roundabout, since lines 6 and 7 undo the power rewrite from line 3. Line 12 is justified as 'derivative of a constant' although line 11 had already evaluated that derivative, which is minor.
gpt-oss:20b: uphold — The step from line 6 to 7 only uses log(exp(u)) = u, so no differentiation or chain rule is applied and the label 'chain' names a rule the step did not use.qwen3.6:27b-mlx: uphold — The objection is correct that line 7 is an algebraic simplification of log(exp(u)) and the derivative is not taken until later, so 'chain' is the wrong label (it should be 'simplify' or 'algebra'). The remarks about step 6 are not defects, because 'rewrite' is acceptable there.agent:pipeline: uphold — Beautiful nomination: The log(2) constant drops out and the square root collapses to the clean result x/(x**2+1). The solution's detour through exp and log in lines 6 and 7 hides that elegance.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-12 — Every line is mathematically correct, but line 7 is labelled and described as a chain-rule step when it is just log(exp(u)) = u, which would teach students that the chain rule is what removes log(exp()). The route is also needlessly roundabout, since lines 6 and 7 undo the power rewrite from line 3. Line 12 is justified as 'derivative of a constant' although line 11 had already evaluated that derivative, which is minor.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 6 applies the identity log(a^b) = b*log(a) but labels it 'rewrite' while Step 7 labels the resulting simplification 'chain'. More critically, Step 7 claims to apply the chain rule to log(exp(...)), but the transition from Step 6 to Step 7 is actually a simplification using log(exp(u)) = u, not a differentiation step. The label 'chain' is incorrect for an algebraic simplification step. Furthermore, Step 6 itself is an algebraic rewrite, but the note in Step 7 is misleading because the step from 6 to 7 is not a differentiation application of the chain rule, but rather a simplification of the logarithm of an exponential. The derivative is not taken in step 7; the expression is simplified. The derivative is taken in step 8. Thus, step 7's label 'chain' is wrong because no differentiation occurs in that step; it is an algebraic simplification.gpt-oss:20b: fail (misleading) 2026-09-26 — Step 7 incorrectly labels the simplification of log(exp(u)) as a chain rule application. The derivative of log(exp(u)) is simply u', so the step should be a rewrite, not a chain rule. This mislabeling could mislead a student into thinking a chain rule is needed here.qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 7 is labeled 'chain' but performs the simplification of log(exp(u)) to u, which is an algebraic/rewrite step, not the application of the chain rule for differentiation. The chain rule is correctly applied in Step 9.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 7 is labeled 'chain' but performs a logarithmic simplification (log(a^b) = b*log(a)), which is an algebraic rewrite, not a differentiation step. The chain rule is applied in Step 9.gpt-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.