Derivative of \( \displaystyle - \frac{3 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{3 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \)
Problem 2.2401 · hard Video
Differentiate \( \displaystyle f(x) = - \frac{3 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{3 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{3 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{3 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2}\right) \]derivative constantStart with the derivative of the function. Distribute the constant factor.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4}\right) + \frac{d}{d x} \frac{3 \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} + \frac{3 \frac{d}{d x} \ln{\left(\tan{\left(2 x + 2 \right)} \right)}}{2} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = - \frac{3 \cos^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(2 x + 2 \right)}}}{4} + \frac{3 \frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} \]chain rewrite algebraApply the chain rule to the logarithmic terms. Rewrite the power using the exponential and logarithm. Simplify the exponent.✓ Proved
- \[ = - \frac{3 \frac{d}{d x} \left(- 2 \ln{\left(\cos{\left(2 x + 2 \right)} \right)}\right)}{4} + \frac{3 \frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} \]chain algebra algebraApply the chain rule to the exponential term. Simplify the fraction by canceling the power term. Simplify the expression.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \ln{\left(\cos{\left(2 x + 2 \right)} \right)}}{2} + \frac{3 \frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} \]constant-multipleCombine the constants.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(2 x + 2 \right)}}{2 \tan{\left(2 x + 2 \right)}} + \frac{3 \frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 \cos{\left(2 x + 2 \right)}} \]chainApply the chain rule to the log terms.✓ Proved
- \[ = - \frac{3 \sin{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} + \frac{3 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]chainDifferentiate the inner trigonometric functions.✓ Proved
- \[ = - 3 \tan{\left(2 x + 2 \right)} + \frac{3 \sec^{2}{\left(2 x + 2 \right)}}{\tan{\left(2 x + 2 \right)}} \]algebra constant-multipleSimplify the terms using tan(u) = sin(u)/cos(u). Distribute the 2 into the parentheses.✓ Proved
- \[ = \frac{3 \left(- \tan^{2}{\left(2 x + 2 \right)} + \sec^{2}{\left(2 x + 2 \right)}\right)}{\tan{\left(2 x + 2 \right)}} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{3}{\tan{\left(2 x + 2 \right)}} \]trig algebraUse the identity sec^2(u) = 1 + tan^2(u). Simplify the numerator.✓ Proved
- \[ = 3 \cot{\left(2 x + 2 \right)} \]rewriteUse the identity cot(u) = 1/tan(u).✓ Proved
Answer \( \frac{3}{\tan{\left(2 x + 2 \right)}} \)
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.
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. 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 undefined where tan(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 undefined where tan(2*x + 2) = 0 undefined where cos(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 undefined where tan(2*x + 2) = 0 undefined where cos(2*x + 2) = 0 |
| 8 | ✓ 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 undefined where cos(2*x + 2) = 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 tan has poles at odd multiples of pi/2 undefined where tan(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 tan(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 tan(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 undefined where cos(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 undefined where cos(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 undefined where cos(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 |
| 16 | ✓ 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 |
| 17 | ✓ 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 |
| 18 | ✓ 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 |
| 19 | ✓ 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 cot has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 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: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-rule constraint, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 5 incorrectly applies the chain rule to the logarithm by writing the derivative as f'(u)/u, which is only valid for log(u) if the outer function is log(u) and the inner is u, but here the argument is a power. The standard chain rule for log(g(x)) is g'(x)/g(x). Step 5 writes Derivative(log(u), x) as Derivative(u, x)/u, which is correct, but it labels the step 'chain' while also implicitly handling the derivative of the log. However, the real error is in Step 6-9 where it unnecessarily rewrites the power as an exponential to differentiate it, but Step 5's application of the chain rule to log(cos(...)**(-2)) is actually correct in form (derivative of outer log is 1/inner, times derivative of inner). Wait, let's look closer. Step 5 says: Derivative(log(u), x) -> Derivative(u, x)/u. This is the chain rule for log. The label 'chain' is acceptable. But Step 6 rewrites u = cos**(-2) as exp(log(u)). This is valid. Step 8 applies chain rule to exp(v). This is valid. Step 9 simplifies exp(log(u))/u to 1. This is valid. Step 10 simplifies to -3/4 * Derivative(-2 log(cos)), which is -3/4 * (-2) * Derivative(log(cos)) = 3/2 Derivative(log(cos)). Step 11 does this constant multiple. So the first term becomes 3/2 * Derivative(log(cos(2x+2))). The second term is 3/2 * Derivative(log(tan(2x+2))). Step 12 combines them? No, Step 12 writes 3/2 * (Derivative(cos)/cos + Derivative(tan)/tan). This implies the first term was 3/2 * Derivative(log(cos)) and the second was 3/2 * Derivative(log(tan)). This is correct. Step 13 differentiates cos and tan. Derivative(cos(2x+2)) is -sin(2x+2)*2. Derivative(tan(2x+2)) is sec^2(2x+2)*2. So we get 3/2 * ( (-2sin)/cos + (2sec^2)/tan ). Step 14 simplifies -2sin/cos to -2tan. So 3/2 * (-2tan + 2sec^2/tan). Step 15 distributes 3/2 * 2 = 3. So 3 * (-tan + sec^2/tan). Step 16 common denominator: 3 * ( (-tan^2 + sec^2)/tan ). Step 17 sec^2 = 1+tan^2. So 3 * ( (-tan^2 + 1 + tan^2)/tan ) = 3 * (1/tan). Step 18 simplifies to 3/tan. Step 19 rewrites to 3cot. The stated answer is 3/tan(2x+2). The solution ends at 3*cot(2x+2). These are equivalent. Is there a defect? Step 5 label 'chain' is fine. Step 6 'rewrite' is fine. Step 8 'chain' is fine. Step 12 'chain' is applied to log(cos) and log(tan). It writes Derivative(log(cos)) as Derivative(cos)/cos. This is correct. The label 'chain' is appropriate. Step 13 'chain' is applied to cos(2x+2) and tan(2x+2). It differentiates the inner linear term. Correct. Step 14 'algebra' simplifies sin/cos to tan. Correct. Step 15 'constant-multiple' distributes 3/2 * 2. Correct. Step 16 'algebra' common denominator. Correct. Step 17 'trig' identity. Correct. Step 18 'algebra' simplify. Correct. Step 19 'rewrite' to cot. Correct. The stated answer is 3/tan(2x+2). The final line is 3*cot(2x+2). The problem asks to differentiate. The solution provides a sequence of steps ending in 3*cot(2x+2). The stated answer is 3/tan(2x+2). These are equal. Is the final step required to match the stated answer exactly? Usually, equivalent forms are accepted. However, look at Step 5 again. It applies the chain rule to log(u). The rule is d/dx log(u) = u'/u. The step writes Derivative(log(u), x) -> Derivative(u, x)/u. This is correct. But wait, Step 5 applies this to BOTH terms simultaneously. Step 4 has two terms. Step 5 transforms both. This violates the 'one thing per step' rule? Step 4: Term1 + Term2. Step 5: TransformedTerm1 + TransformedTerm2. This applies the chain rule to two separate terms in one step. This is a defect. A step should change ONE thing. Changing two independent terms is applying the rule twice. Therefore, it should be split into two steps. This is a 'style' or 'error' depending on strictness. The prompt says 'A step that applies two rules at once is a defect.' Applying the same rule to two separate terms is often considered applying the rule twice. Let's look for a more severe error. Step 12: 3/2 * Derivative(log(cos)) + 3/2 * Derivative(log(tan)) -> 3/2 * (Derivative(cos)/cos + Derivative(tan)/tan). This applies the chain rule to log(cos) AND log(tan) in one step. Again, two changes. Step 13: Differentiates cos(2x+2) and tan(2x+2) in one step. Two changes. The solution consistently applies rules to multiple terms in a single step. This violates the 'Each step changes ONE thing' contract. Specifically, Step 5 changes two terms. Step 12 changes two terms. Step 13 changes two terms. This is a defect. I will flag Step 5 as the first instance.gpt-oss:20b: fail (error) 2026-10-15 — Step 6 applies both a rewrite (exp(log(...))) and a chain rule in one line, violating the rule that each step must change only one thing.
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 2026-10-18 with SymPy 1.14.0.