Derivative of \( \displaystyle - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{2} + \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \)
Problem 2.2420 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{2} + \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{2} + \ln{\left(\tan{\left(2 x + 2 \right)} \right)}\right) \]sumStart with the derivative of the sum.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{2}\right) + \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \]constant-multipleApply the constant multiple rule to the first term.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{2} + \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = - \frac{\cos^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(2 x + 2 \right)}}}{2} + \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \]powerApply the power rule to the inner function.✓ Proved
- \[ = \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} + \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} \]chainApply the chain rule to the power function.✓ Proved
- \[ = - \frac{\sin{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{\cos{\left(2 x + 2 \right)}} + \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \]derivativeApply the derivative rule for the cosine function.✓ Proved
- \[ = - \frac{2 \sin{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} + \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \]algebra algebra algebra algebraDifferentiate the innermost linear function. Simplify the numerator. Simplify the fraction using exponent rules. Distribute the -1/2.✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)} \]algebraRewrite sin/cos as tan.✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + \frac{\frac{d}{d x} \tan{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + \frac{\sec^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{\tan{\left(2 x + 2 \right)}} \]trigApply the derivative rule for the tangent function.✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + \frac{2 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]algebraDifferentiate the innermost linear function.✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + \frac{2}{\sin{\left(2 x + 2 \right)} \cos{\left(2 x + 2 \right)}} \]algebra algebraRewrite sec^2/tan as 1/(sin*cos). Express the fraction with a common denominator structure.✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + \frac{4}{\sin{\left(4 x + 4 \right)}} \]algebraUse the double angle identity 2sin(u)cos(u) = sin(2u).✓ Proved
- \[ = - 2 \tan{\left(2 x + 2 \right)} + 4 \csc{\left(4 x + 4 \right)} \]trig simplifyConvert 1/sin to csc. Final simplified form.✓ Proved
Answer \( \frac{2}{\tan{\left(2 x + 2 \right)}} \)
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.
Lines: 19 proved, 1 checked numerically. 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. 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 tan has poles at odd multiples of pi/2 undefined where cos(2*x + 2) = 0 |
| 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 undefined where cos(2*x + 2) = 0 |
| 4 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 5 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 6 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 7 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 8 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 9 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 10 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 11 | ✓ 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 undefined where cos(2*x + 2) = 0 |
| 12 | ✓ 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 undefined where tan(2*x + 2) = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(2*x + 2) = 0 sec has poles at odd multiples of pi/2 |
| 14 | ✓ 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) = 0 |
| 15 | ✓ 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) = 0 undefined where cos(2*x + 2) = 0 undefined where sin(2*x + 2) = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(2*x + 2) = 0 undefined where sin(2*x + 2) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(2*x + 2) = 0 undefined where sin(2*x + 2) = 0 undefined where sin(4*x + 4) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(4*x + 4) = 0 csc has poles at multiples of pi |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 csc has poles at multiples of pi |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -2*tan(2*x + 2) - 2/tan(2*x + 2) + 4/sin(4*x + 4); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(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: passqwen3.6:27b-mlx: fail (style) — Step 2 is labeled 'constant-multiple' but the step actually splits the derivative of a sum into the sum of derivatives (the 'sum' rule); the constant multiple rule is applied in the subsequent step 3. Step 13 is labeled 'trig' but applies the derivative of tangent (chain rule + derivative rule), which should be labeled 'derivative' or 'chain'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-19 — Step 2 is labeled 'constant-multiple' but the step actually splits the derivative of a sum into the sum of derivatives (the 'sum' rule); the constant multiple rule is applied in the subsequent step 3. Step 13 is labeled 'trig' but applies the derivative of tangent (chain rule + derivative rule), which should be labeled 'derivative' or 'chain'.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 2 is labeled 'constant-multiple' but performs the 'sum' rule (splitting the derivative of a sum into a sum of derivatives). The constant multiple rule was not applied in this step; it was applied in Step 3.gpt-oss:20b: fail (style) 2026-10-16 — Step 7 is labeled "algebra" but it applies the derivative rule to the linear function 2*x+2. The correct label should be "derivative".qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 2 is labeled 'constant-multiple' but performs the 'sum' rule (splitting the derivative of a sum into a sum of derivatives). The constant multiple rule was not applied in this step; it was applied in step 3. This violates the contract that each step must name the rule it actually applies.gpt-oss:20b: fail (error) 2026-10-16 — Step 16 changes the expression from 2/(sin(2*x+2)*cos(2*x+2)) to 4/(2*sin(2*x+2)*cos(2*x+2)) in a single step, which applies two algebraic manipulations at once (scaling numerator and denominator).
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 2026-10-19 with SymPy 1.14.0.