Derivative of \( \displaystyle \tan{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} \)
Problem 2.363 · hard Video
Differentiate \( \displaystyle f(x) = \tan{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} \).
- \[ \frac{d}{d x} \tan{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \tan{\left(5 x + 1 \right)} \frac{d}{d x} \sec^{2}{\left(5 x + 1 \right)} + \sec^{2}{\left(5 x + 1 \right)} \frac{d}{d x} \tan{\left(5 x + 1 \right)} \]productApply the product rule.✓ Proved
- \[ = 2 \tan{\left(5 x + 1 \right)} \sec{\left(5 x + 1 \right)} \frac{d}{d x} \sec{\left(5 x + 1 \right)} + \sec^{2}{\left(5 x + 1 \right)} \frac{d}{d x} \tan{\left(5 x + 1 \right)} \]powerApply the power rule to the second term.✓ Proved
- \[ = 2 \tan{\left(5 x + 1 \right)} \sec{\left(5 x + 1 \right)} \frac{d}{d x} \sec{\left(5 x + 1 \right)} + 5 \sec^{4}{\left(5 x + 1 \right)} \]chain algebraDifferentiate the tangent term using the chain rule. Simplify the first term.✓ Checked independently
- \[ = 10 \tan^{2}{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} + 5 \sec^{4}{\left(5 x + 1 \right)} \]chain algebraDifferentiate the secant term using the chain rule. Multiply the terms together.✓ Proved
- \[ = 5 \left(2 \tan^{2}{\left(5 x + 1 \right)} + \sec^{2}{\left(5 x + 1 \right)}\right) \sec^{2}{\left(5 x + 1 \right)} \]algebraFactor out the common term.✓ Proved
- \[ = 5 \left(3 \tan^{2}{\left(5 x + 1 \right)} + 1\right) \sec^{2}{\left(5 x + 1 \right)} \]algebra simplifyUse the identity sec(u)^2 = 1 + tan(u)^2. Combine like terms.≈ Checked numerically
Answer \( \frac{5 \left(2 - \cos{\left(10 x + 2 \right)}\right)}{\cos^{4}{\left(5 x + 1 \right)}} \)
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.
✓ Nihil obstat Lines: 8 proved, 1 checked by independent solution, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 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 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 4 | ✓ Checked independently | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left 5*(tan(5*x + 1)**2 - sec(5*x + 1)**2 + 1)*sec(5*x + 1)**2; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: d/dx tan(u)=5 sec^2(u) verified; substituting it into line 3 gives line 4 ( checks/2.363/line4-82aacf35.py)tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 5 | ✓ 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 |
| 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 |
| 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 |
| 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 |
| 9 | ≈ Checked numerically | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: simplify left 5*(-tan(5*x + 1)**2 + sec(5*x + 1)**2 - 1)*sec(5*x + 1)**2; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: script refused before running: uses S ( checks/2.363/line9-db759718.py)tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 10 | ✓ 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 |
| answer | ≈ Checked numerically | claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left 5*((3*tan(5*x + 1)**2 + 1)*cos(5*x + 1)**4*sec(5*x + 1)**2 + cos(10*x + 2) - 2)/cos(5*x + 1)**4; numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: script refused before running: uses S ( checks/2.363/line11-050c2eb4.py)undefined where cos(5*x + 1) = 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 4 applies the chain rule to the tangent term but leaves the derivative of the secant term untouched, violating the 'one change per step' rule; Step 6 then applies the chain rule to the secant term. These should be separate steps or labeled differently to reflect that only one sub-expression is being differentiated at a time.
Senior review claude-sonnet-5-5, 2026-10-07: fail — The math is correct from start to finish, and the result equals the stated answer, since 2 - cos(2u) = 3 - 2cos^2 u. Step 9 substitutes the identity sec^2 = 1 + tan^2 but is labeled 'algebra' where 'trig' would be the accurate rule name.
qwen3.6:27b-mlx: dismiss — Differentiating the tangent factor by the chain rule while leaving the secant term untouched is one rule applied to one term, which the contract counts as a single change.
Every verdict on record (11)
claude-sonnet-5-5: fail (style) 2026-10-07 — The math is correct from start to finish, and the result equals the stated answer, since 2 - cos(2u) = 3 - 2cos^2 u. Step 9 substitutes the identity sec^2 = 1 + tan^2 but is labeled 'algebra' where 'trig' would be the accurate rule name.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 applies the chain rule to the tangent term but leaves the derivative of the secant term untouched, violating the 'one change per step' rule; Step 6 then applies the chain rule to the secant term. These should be separate steps or labeled differently to reflect that only one sub-expression is being differentiated at a time.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 4 applies the chain rule to differentiate tan(5x+1) but leaves the second term's derivative (Derivative(sec(5x+1), x)) unevaluated, violating the 'one thing per step' rule by mixing differentiation of one term with simplification of another. Furthermore, the final answer in the stated answer does not match the result of the steps (which simplifies to 5*sec(5x+1)^2*(1+3*tan(5x+1)^2)), indicating a mismatch between the derivation and the claimed final result.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer is mathematically incorrect; the derivative of tan(u)sec(u)^2 is 5sec(u)^2(1+3tan(u)^2), which does not simplify to 5(2-cos(2u))/cos(u)^4. Additionally, Step 4 applies the chain rule to the first term but leaves the derivative operator on the second term, violating the 'one rule per step' constraint by performing partial differentiation.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the product, chain, and power rules in separate steps. The algebraic simplifications and trigonometric identities are applied correctly, and the final answer matches the stated result.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.