Derivative of \( \displaystyle \frac{\tan{\left(x \right)}}{x} \)
Problem 2.417 · hard
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(x \right)}}{x} \).
- \[ \frac{d}{d x} \frac{\tan{\left(x \right)}}{x} \]derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product.✓ Proved
- \[ = \tan{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\frac{d}{d x} \tan{\left(x \right)}}{x} \]productApply the product rule.✓ Proved
- \[ = \tan{\left(x \right)} \frac{d}{d x} \frac{1}{x} + \frac{\sec^{2}{\left(x \right)}}{x} \]trigDifferentiate the first part of the product.✓ Proved
- \[ = \frac{\sec^{2}{\left(x \right)}}{x} - \frac{\tan{\left(x \right)}}{x^{2}} \]derivative algebraDifferentiate the second part of the product. Simplify the terms.✓ Proved
- \[ = \frac{x \sec^{2}{\left(x \right)} - \tan{\left(x \right)}}{x^{2}} \]simplifyCombine into a single fraction.✓ Proved
Answer \( \frac{\frac{x}{\cos^{2}{\left(x \right)}} - \tan{\left(x \right)}}{x^{2}} \)
✓ Nihil obstat 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. 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 tan has poles at odd multiples of pi/2 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where x = 0 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 undefined where x = 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 x = 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 x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where cos(x) = 0 undefined where x = 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 — The solution correctly applies the product rule and standard derivatives, with each step changing only one aspect of the expression. The final answer is algebraically equivalent to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule and standard derivatives, with each step changing only one aspect of the expression. The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule and standard derivatives. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule and standard derivative rules. Each step modifies only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20
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.