Derivative of \( \displaystyle \ln{\left(\frac{x}{x + 1} \right)} \)
Problem 2.8 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{x}{x + 1} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\frac{x}{x + 1} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \ln{\left(x + 1 \right)}\right) \]logarithmicUse the property log(a/b) = log(a) - log(b).✓ Proved
- \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \ln{\left(x + 1 \right)} \]sumDistribute the derivative across the subtraction.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(x + 1 \right)} + \frac{1}{x} \]derivativeDifferentiate the first term.✓ Proved
- \[ = - \frac{1}{x + 1} + \frac{1}{x} \]derivative rewrite simplifyDifferentiate the second term using the chain rule. Rewrite 1/x with a common denominator. Simplify the expression.✓ Proved
- \[ = \frac{1}{x \left(x + 1\right)} \]algebra simplifyCombine the fractions using a common denominator. Simplify the numerator.✓ Proved
Answer \( \frac{1}{x \left(x + 1\right)} \)
Mind the domain. The answer is also defined on (-1, 0), where f(x) is not. Substituting there gives a number that is not a slope of f.
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 log is undefined for non-positive arguments undefined where x + 1 = 0 |
| 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 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
deepseek-r1:70b: passgpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — Step 5 claims to use the 'derivative' rule but the note explicitly mentions the 'chain rule'; the label 'derivative' is reserved for basic forms (like d/dx of log(x)), whereas differentiating log(x+1) requires the 'chain' rule. Additionally, steps 6-9 are logically redundant and circular, as step 7 reverts to the expression in step 5, but the primary defect is the incorrect labeling of the chain rule application.
Every verdict on record (11)
qwen3.6:27b-mlx: fail (misleading) 2026-09-20 — Step 5 claims to use the 'derivative' rule but the note explicitly mentions the 'chain rule'; the label 'derivative' is reserved for basic forms (like d/dx of log(x)), whereas differentiating log(x+1) requires the 'chain' rule. Additionally, steps 6-9 are logically redundant and circular, as step 7 reverts to the expression in step 5, but the primary defect is the incorrect labeling of the chain rule application.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (misleading) 2026-09-20 — Step 5 is labeled 'derivative' but applies the chain rule to differentiate log(x+1); the label should be 'chain'. Additionally, steps 6-9 are logically incoherent: step 6 rewrites 1/x, step 7 reverts to the previous expression, and step 8 combines fractions, making the sequence confusing and pedagogically unsound.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (misleading) 2026-09-19 — Step 5 is labeled 'derivative' but applies the chain rule to differentiate log(x+1); the label should be 'chain'. Additionally, Step 7 undoes the work of Step 6, creating a redundant and confusing loop in the simplification process.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 is labeled 'derivative' but applies the chain rule to differentiate log(x+1); the label should be 'chain'. Additionally, steps 6-9 are logically incoherent: step 6 rewrites 1/x, step 7 reverts to the previous expression, and step 8 combines fractions, creating a non-linear and confusing derivation path.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 5 incorrectly labels the application of the chain rule as 'derivative'. This could mislead a student into thinking the derivative of log(x + 1) is 1/(x + 1) without considering the chain rule.gpt-oss:20b: pass 2026-09-19deepseek-r1:70b: pass 2026-09-16
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.