Derivative of \( \displaystyle \frac{\tan{\left(3 x \right)}}{3} \)
Problem 2.1169 · easy
Differentiate \( \displaystyle f(x) = \frac{\tan{\left(3 x \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\tan{\left(3 x \right)}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(3 x \right)}}{3} \]constant-multiple rewrite constant-multiplePull out the constant factor. Rewrite the tangent function using the exponential identity. The constant is already factored, but we ensure the structure is clear.✓ Proved
- \[ = \frac{\tan{\left(3 x \right)} \frac{d}{d x} \ln{\left(\tan{\left(3 x \right)} \right)}}{3} \]chain rewriteApply the chain rule to the exponential function. Simplify the exponential term back to its original form.✓ Proved
- \[ = \frac{\frac{d}{d x} \tan{\left(3 x \right)}}{3} \]chain algebraApply the chain rule to the logarithm. Cancel the tan(3*x) terms.✓ Proved
- \[ = \sec^{2}{\left(3 x \right)} \]derivative simplifyDifferentiate the tangent function using the chain rule. Simplify the final expression.✓ Proved
Answer \( \tan^{2}{\left(3 x \right)} + 1 \)
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.
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. 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan 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 log is undefined for non-positive arguments |
| 6 | ✓ 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 |
| 7 | ✓ 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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 9 | ✓ 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 |
| 10 | ✓ 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 tan has poles at odd multiples of pi/2 |
| 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 9 applies the chain rule to differentiate tan(3*x), but is labeled 'derivative'. The label 'derivative' is reserved for unfolding d/dx on a known form (like d/dx sin(x)), whereas applying the chain rule requires the label 'chain'. Additionally, Step 3 introduces an unnecessary and complex rewrite using exp(log(...)) which obscures the standard differentiation path, though the primary defect is the mislabeling in Step 9.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 9 applies the chain rule to differentiate tan(3*x), but is labeled 'derivative'. The label 'derivative' is reserved for unfolding d/dx on a known form (like d/dx sin(x)), whereas applying the chain rule requires the label 'chain'. Additionally, Step 3 introduces an unnecessary and complex rewrite using exp(log(...)) which obscures the standard differentiation path, though the primary defect is the mislabeling in Step 9.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 4 is a no-op that violates the 'one change' rule by applying 'constant-multiple' to an expression where the constant is already factored out, serving no logical purpose. Step 9 is labeled 'derivative' but performs both the differentiation of tan and the chain rule multiplication by 3, violating the single-rule constraint.gpt-oss:20b: fail (style) 2026-09-29 — [domain objection, downgraded to style] Step 3 rewrites tan(3*x) as exp(log(tan(3*x))), which is not an identity for all real x (tan can be negative, making log undefined). This introduces a false transformation that invalidates the subsequent steps.
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 2026-09-29 with SymPy 1.14.0.