Derivative of \( \displaystyle \tan{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} \)
Problem 2.363 · hard
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 numerically
- \[ = 10 \tan^{2}{\left(5 x + 1 \right)} \sec^{2}{\left(5 x + 1 \right)} + 5 \sec^{4}{\left(5 x + 1 \right)} \]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.
Lines: 8 proved, 3 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.
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 numerically | 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 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 | 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 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 | 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 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.
Every verdict on record (10)
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.