Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)} \right)}}{5} \)
Problem 2.957 · hard Video
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)} \right)}}{5} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)} \right)}}{5} \]constant-multiplePull out the constant factor 1/5.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)} \right)}}{5} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)}\right)}{5 \left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)}\right)} \]sumDifferentiate the sum inside the argument.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(5 x - 3 \right)} + \frac{d}{d x} \sec{\left(5 x - 3 \right)}}{5 \left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)}\right)} \]chainApply the chain rule to each trigonometric term.✓ Proved
- \[ = \frac{5 \tan{\left(5 x - 3 \right)} \sec{\left(5 x - 3 \right)} + 5 \sec^{2}{\left(5 x - 3 \right)}}{5 \left(\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)}\right)} \]constant-multipleDifferentiate the inner functions and factor out 5.≈ Checked numerically
- \[ = \frac{\tan{\left(5 x - 3 \right)} \sec{\left(5 x - 3 \right)} + \sec^{2}{\left(5 x - 3 \right)}}{\tan{\left(5 x - 3 \right)} + \sec{\left(5 x - 3 \right)}} \]algebra algebraFactor out the constant 5 from the sum. Simplify the expression by canceling 1/5 and 5.✓ Proved
- \[ = \sec{\left(5 x - 3 \right)} \]algebra simplifyFactor out sec(5*x - 3) from the numerator. Cancel the common factor in the numerator and denominator.✓ Proved
Answer \( \frac{1}{\cos{\left(5 x - 3 \right)}} \)
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: 9 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 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(5*x - 3) + sec(5*x - 3) = 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(5*x - 3) + sec(5*x - 3) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (tan(5*x - 3)**2 - sec(5*x - 3)**2 + 1)/(tan(5*x - 3) + sec(5*x - 3)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(5*x - 3) + sec(5*x - 3) = 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(5*x - 3) + sec(5*x - 3) = 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(5*x - 3) + sec(5*x - 3) = 0 |
| 8 | ✓ 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(5*x - 3) + sec(5*x - 3) = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 - 3) = 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 (error) — Step 5 is labeled 'constant-multiple' but performs the differentiation of tan and sec (applying chain rule and known derivatives), which is not what the label describes. The label should be 'derivative' or 'chain'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 5 is labeled 'constant-multiple' but performs the differentiation of tan and sec (applying chain rule and known derivatives), which is not what the label describes. The label should be 'derivative' or 'chain'.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied (e.g., constant-multiple for factoring constants, chain for composite functions, sum for linearity of differentiation, and algebra/simplify for rearranging and canceling terms).gpt-oss:20b: fail (style) 2026-09-27 — Step 4 incorrectly labels the sum of derivatives as a chain rule application, and step 5 labels a combination of chain and constant‑multiple rules only as constant‑multiple. These mislabelings violate the one‑rule‑per‑step rule and could mislead a student about which rule is actually used.qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 5 applies the chain rule to differentiate the inner trigonometric functions, but is labeled 'constant-multiple'. Step 6 is labeled 'algebra' for factoring, which is acceptable, but Step 5's label is incorrect for the differentiation performed.gpt-oss:20b: fail (style) 2026-09-27 — Step 4 applies the sum rule, not the chain rule. The label "chain" is incorrect for that step.
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 2026-09-28 with SymPy 1.14.0.