Derivative of \( \displaystyle \frac{\ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)}}{5} \)
Problem 2.1657 · hard Beautiful
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \frac{\ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)}}{5} \]constant-multipleMove the constant outside the derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)} \right)}}{5} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}\right)}{5 \left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}\right)} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(- \cot{\left(5 x \right)}\right) + \frac{d}{d x} \csc{\left(5 x \right)}}{5 \left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}\right)} \]constant-multipleDistribute the negative sign.✓ Proved
- \[ = \frac{- \frac{d}{d x} \cot{\left(5 x \right)} + \frac{d}{d x} \csc{\left(5 x \right)}}{5 \left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}\right)} \]trigDifferentiate the trigonometric functions.✓ Proved
- \[ = \frac{- 5 \cot{\left(5 x \right)} \csc{\left(5 x \right)} + 5 \csc^{2}{\left(5 x \right)}}{5 \left(- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}\right)} \]chain algebraApply the chain rule to cot(5x) and csc(5x). Simplify the expression inside the parentheses.✓ Proved
- \[ = \frac{- \cot{\left(5 x \right)} \csc{\left(5 x \right)} + \csc^{2}{\left(5 x \right)}}{- \cot{\left(5 x \right)} + \csc{\left(5 x \right)}} \]algebra algebraFactor out the common 5. Cancel the 1/5 and 5.✓ Proved
- \[ = \csc{\left(5 x \right)} \]algebra simplifyFactor csc(5x) out of the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\sin{\left(5 x \right)}} \)
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 csc has poles at multiples of pi cot has poles at multiples of pi |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi cot has poles at multiples of pi undefined where -cot(5*x) + csc(5*x) = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 csc has poles at multiples of pi |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(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: fail (style) — Step 4 applies the sum rule to split the derivative of a sum, but it is labeled "constant-multiple". The label should be "sum" (or "algebra" if viewed as a distributive step).qwen3.6:27b-mlx: fail (error) — Step 6 is labeled 'chain' but applies the derivative rules for cotangent and cosecant (which are 'derivative' or 'trig' rules) while simultaneously handling the inner derivative of 5x. The label 'chain' does not name the differentiation of the trigonometric functions themselves, and the step combines multiple differentiation rules (derivative of cot, derivative of csc, and chain rule for the inner 5x) into a single step, violating the one-rule-per-step constraint.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-06 — Step 4 applies the sum rule to split the derivative of a sum, but it is labeled "constant-multiple". The label should be "sum" (or "algebra" if viewed as a distributive step).qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 6 is labeled 'chain' but applies the derivative rules for cotangent and cosecant (which are 'derivative' or 'trig' rules) while simultaneously handling the inner derivative of 5x. The label 'chain' does not name the differentiation of the trigonometric functions themselves, and the step combines multiple differentiation rules (derivative of cot, derivative of csc, and chain rule for the inner 5x) into a single step, violating the one-rule-per-step constraint.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 6 is labeled 'chain' but performs the differentiation of cot(5x) and csc(5x) using their standard derivatives (which is 'derivative' or 'trig') combined with the chain rule. More critically, Step 5 is labeled 'trig' but does not perform differentiation; it merely sets up the derivatives. The actual differentiation happens in Step 6. However, the primary defect is that Step 6 applies the chain rule AND the specific trig derivatives simultaneously, violating the 'one rule per step' constraint. It should be split into applying the chain rule to get 5*Derivative(cot(u), u) and then applying the trig derivative rule. Alternatively, if 'trig' covers the derivative of trig functions, Step 6 mixes 'trig' and 'chain'. A step cannot be both. Let's look closer. Step 5 says 'Differentiate the trigonometric functions' but the expression still has 'Derivative(...)'. Step 6 replaces 'Derivative(cot(5*x), x)' with '-csc(5*x)**2 * 5'. This is applying the chain rule AND the derivative of cot. This is two rules. Also, Step 2 is labeled 'chain' but the expression is just `1/5 * Derivative(...)`. The chain rule application happens in Step 3. Step 2 is just a placeholder or mislabeled step. Actually, Step 2 to 3 is the chain rule. Step 2 is just rewriting the constant multiple? No, Step 1 to 2 is constant multiple. Step 2 to 3 is chain. So Step 2 label 'chain' is wrong because the chain rule is applied in the transition to Step 3. Step 2 is just the result of Step 1. It doesn't apply a rule itself, it's the state. Wait, the steps are transitions. Step 1 -> 2 is constant-multiple. Step 2 -> 3 is chain. Step 2 is labeled 'chain'. This is a labeling error: the label should describe the transition FROM the previous step. Step 2 is the result of applying constant-multiple to Step 1. So Step 2's label should be constant-multiple? No, the label is on the step line. Usually, the label describes the rule used to get TO that line. Step 1 is the problem statement. Step 2 is obtained by constant-multiple. So Step 2's label should be constant-multiple? But Step 2 is labeled 'chain'. This is a defect. Step 3 is obtained by chain rule from Step 2. Step 3 is labeled 'sum'. This is a defect. Step 4 is obtained by sum rule from Step 3. Step 4 is labeled 'constant-multiple'. This is a defect. Step 5 is obtained by constant-multiple (pulling out -1) from Step 4. Step 5 is labeled 'trig'. This is a defect. Step 6 is obtained by derivative/chain/trig from Step 5. Step 6 is labeled 'chain'. This is a defect (mixed rules). Step 7 is algebra. Step 8 is algebra. Step 9 is algebra. Step 10 is algebra. Step 11 is simplify. The labels are completely out of sync with the transitions. Step N's label should describe the rule used to transform Step N-1 into Step N. Step 1 -> 2: constant-multiple. Label on 2 is 'chain'. Wrong. Step 2 -> 3: chain. Label on 3 is 'sum'. Wrong. Step 3 -> 4: sum. Label on 4 is 'constant-multiple'. Wrong. Step 4 -> 5: constant-multiple (distributing negative). Label on 5 is 'trig'. Wrong. Step 5 -> 6: derivative/chain/trig. Label on 6 is 'chain'. Partially correct but mixes rules. This is a systematic labeling error.gpt-oss:20b: fail (style) 2026-10-06 — Step 4 incorrectly labels the sum rule as "constant-multiple", and step 5 incorrectly labels the application of the constant‑multiple rule as "trig". Both steps apply a single rule but the labels do not match the rule used.
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 2026-10-06 with SymPy 1.14.0.