Derivative of \( \displaystyle 5 \ln{\left(\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)} \right)} \)
Problem 2.1349 · hard Beautiful
Differentiate \( \displaystyle f(x) = 5 \ln{\left(\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)} \right)} \).
- \[ \frac{d}{d x} 5 \ln{\left(\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 5 \frac{d}{d x} \ln{\left(\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)} \right)} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)}\right)}{\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)}} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} \tan{\left(x + 2 \right)} + \frac{d}{d x} \sec{\left(x + 2 \right)}\right)}{\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{5 \left(\tan{\left(x + 2 \right)} \sec{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right) + \sec^{2}{\left(x + 2 \right)} \frac{d}{d x} \left(x + 2\right)\right)}{\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)}} \]trigApply the derivatives of tangent and secant functions.≈ Checked numerically
- \[ = \frac{5 \left(\tan{\left(x + 2 \right)} \sec{\left(x + 2 \right)} + \sec^{2}{\left(x + 2 \right)}\right)}{\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)}} \]derivativeDifferentiate the inner linear functions.✓ Proved
- \[ = \frac{5 \tan{\left(x + 2 \right)} \sec{\left(x + 2 \right)} + 5 \sec^{2}{\left(x + 2 \right)}}{\tan{\left(x + 2 \right)} + \sec{\left(x + 2 \right)}} \]algebraMultiply the terms together.✓ Proved
- \[ = 5 \sec{\left(x + 2 \right)} \]algebra simplifyFactor out the common term sec(x + 2). Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\cos{\left(x + 2 \right)}} \)
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: 9 proved, 1 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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(x + 2) + sec(x + 2) = 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(x + 2) + sec(x + 2) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 5*(tan(x + 2)**2 - sec(x + 2)**2 + 1)/(tan(x + 2) + sec(x + 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 undefined where tan(x + 2) + sec(x + 2) = 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(x + 2) + sec(x + 2) = 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(x + 2) + sec(x + 2) = 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(x + 2) + sec(x + 2) = 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(x + 2) = 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: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30gpt-oss:20b: fail (error) 2026-09-30 — Step 5 applies two differentiation rules at once (the derivatives of tan and sec) in a single line, violating the rule that each step must change only one thing. It should be split into two separate steps, each labeled "derivative" (or "trig" for the specific function).
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 2026-10-03 with SymPy 1.14.0.