Derivative of \( \displaystyle \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \)
Problem 2.368 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \).
- \[ \frac{d}{d x} \left(\frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{\ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{d}{d x} \frac{\ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \]sumApply the difference rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} + 1 \right)}}{10} \]constant algebraFactor out the constant 1/10. Factor out 1/10 from the expression.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{\frac{d}{d x} \left(\sin{\left(5 x \right)} + 1\right)}{10 \left(\sin{\left(5 x \right)} + 1\right)} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\sin{\left(5 x \right)} - 1 \right)}}{10} - \frac{\cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right)} \]chain algebraApply the chain rule to the inner function sin(5*x) + 1. Simplify the fraction.✓ Proved
- \[ = - \frac{\cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right)} + \frac{\cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} - 1\right)} \]chain algebraApply the chain rule to the first term. Simplify the first term's fraction.✓ Proved
- \[ = \frac{\left(- \frac{1}{\sin{\left(5 x \right)} + 1} + \frac{1}{\sin{\left(5 x \right)} - 1}\right) \cos{\left(5 x \right)}}{2} \]constant-multipleFactor out 5 * cos(5*x) and adjust the constant.✓ Proved
- \[ = \frac{\cos{\left(5 x \right)}}{\left(\sin{\left(5 x \right)} - 1\right) \left(\sin{\left(5 x \right)} + 1\right)} \]algebra simplify algebraCombine the fractions using a common denominator. Simplify the numerator. Cancel the 2 and the 1/2.✓ Proved
- \[ = \frac{\cos{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)} - 1} \]algebraUse the difference of squares formula.✓ Proved
- \[ = - \frac{\cos{\left(5 x \right)}}{1 - \sin^{2}{\left(5 x \right)}} \]algebraRewrite the denominator.✓ Proved
- \[ = - \frac{1}{\cos{\left(5 x \right)}} \]algebra simplifyUse the identity 1 - sin(x)^2 = cos(x)^2. Simplify the expression.✓ Proved
- \[ = - \sec{\left(5 x \right)} \]rewriteRewrite 1/cos(x) as sec(x).✓ Proved
Answer \( - \frac{1}{\cos{\left(5 x \right)}} \)
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 |
| 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 undefined where sin(5*x) + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x) + 1 = 0 undefined where sin(5*x) - 1 = 0 undefined where sin(5*x)**2 - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(5*x)**2 - 1 = 0 undefined where 1 - sin(5*x)**2 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - sin(5*x)**2 = 0 undefined where cos(5*x) = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(5*x) = 0 sec has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(5*x) = 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 3 is labeled 'constant' but performs the extraction of a constant multiple, which should be labeled 'constant-multiple'. Step 10 is labeled 'constant-multiple' but performs factoring of a variable term (cos(5*x)), which is an algebraic manipulation and should be labeled 'algebra'.
Every verdict on record (10)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 is labeled 'constant' but performs the extraction of a constant multiple, which should be labeled 'constant-multiple'. Step 10 is labeled 'constant-multiple' but performs factoring of a variable term (cos(5*x)), which is an algebraic manipulation and should be labeled 'algebra'.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 incorrectly labels the extraction of the constant factor 1/10 as 'constant'. The rule 'constant' typically refers to the derivative of a constant being zero. Extracting a constant multiplier from a derivative requires the 'constant-multiple' rule. Additionally, Step 5 applies the chain rule to only one term of a difference, which violates the 'one change per step' principle if interpreted as a single operation on the whole expression, but more critically, the label 'chain' is applied to a step that also implicitly relies on the derivative of log, making it a compound step if not carefully separated, though the primary defect is the mislabeling in Step 3.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels used are from the allowed vocabulary and accurately describe the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 3 is labeled 'constant' but performs the extraction of a constant factor, which should be labeled 'constant-multiple'. Step 10 is labeled 'constant-multiple' but performs factoring of a variable term (cos(5*x)), which is an algebraic manipulation and should be labeled 'algebra'.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19
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.