Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \)
Problem 2.126 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \]constant-multipleBring the constant factor out of the derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)} \right)}}{2} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)}\right)}{2 \left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)}\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(2 x + 2 \right)} + \frac{d}{d x} \sec{\left(2 x + 2 \right)}}{2 \left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)}\right)} \]chainApply the chain rule to each term in the sum.✓ Proved
- \[ = \frac{2 \tan{\left(2 x + 2 \right)} \sec{\left(2 x + 2 \right)} + 2 \sec^{2}{\left(2 x + 2 \right)}}{2 \left(\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)}\right)} \]trigDifferentiate the tangent and secant functions using the chain rule.✓ Proved
- \[ = \frac{\tan{\left(2 x + 2 \right)} \sec{\left(2 x + 2 \right)} + \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)} + \sec{\left(2 x + 2 \right)}} \]algebra simplifyFactor out the common factor of 2. Cancel the 1/2 and the 2, then simplify the fraction.✓ Proved
Answer \( \frac{1}{\cos{\left(2 x + 2 \right)}} \)
Mind the domain. The answer is also defined at points 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 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x + 2) + sec(2*x + 2) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x + 2) + sec(2*x + 2) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x + 2) + sec(2*x + 2) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x + 2) + sec(2*x + 2) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(2*x + 2) + sec(2*x + 2) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(2*x + 2) = 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 (style) — Step 5 applies the chain rule to differentiate tan and sec, but is labeled 'trig'; the label should be 'chain' or 'derivative' to reflect the differentiation rule applied. Step 7 combines cancellation and algebraic simplification into one step, violating the one-change-per-step constraint.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 5 applies the chain rule to differentiate tan and sec, but is labeled 'trig'; the label should be 'chain' or 'derivative' to reflect the differentiation rule applied. Step 7 combines cancellation and algebraic simplification into one step, violating the one-change-per-step constraint.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 5 applies the chain rule to differentiate tan(2x+2) and sec(2x+2), but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for differentiation. Additionally, Step 7 combines cancellation and algebraic simplification into one step, violating the one-change-per-step rule.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the operation as "sum"; it is simply a product of 1/2, 1/(tan+sec), and the derivative of the inner sum. The correct label would be "product" or simply "algebra".qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 5 applies the chain rule to differentiate tan(2x+2) and sec(2x+2) but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for differentiation. Furthermore, Step 5 combines the differentiation of two separate terms (which should be handled by 'sum' or individual 'chain' steps) into a single step, violating the one-rule-per-step constraint.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (style) 2026-09-19 — Step 4 incorrectly labels the application of the sum rule as "chain". The step differentiates a sum of two functions, so the correct label should be "sum" rather than "chain".qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 4 applies both the sum rule and the chain rule in one line, violating the one‑rule‑per‑step rule. Step 5 also combines the chain rule with the trig derivative in a single step, which is not allowed. The labels used do not accurately reflect the rules applied.qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: fail (style) 2026-09-18 — Step 5 applies the chain rule to both the tangent and secant terms in a single step, changing two terms at once. Each step should modify only one part of the expression.gpt-oss:20b: fail 2026-09-17 — Step 3 mislabels the rule: it applies both the chain rule for the logarithm and the sum rule for the inner derivative, yet it is marked only as 'sum', which could mislead a student.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.