Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.630

Derivative of \( \displaystyle \frac{\tan{\left(2 x \right)}}{2} \)

Problem 2.630 · easy Video

Differentiate \( \displaystyle f(x) = \frac{\tan{\left(2 x \right)}}{2} \).

\[ \frac{d}{d x} \frac{\tan{\left(2 x \right)}}{2} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \tan{\left(2 x \right)}}{2} \]
constant-multiple rewritePull out the constant factor. Note: The previous error was due to improper chain rule notation. We will use the chain rule directly on the tangent function.✓ Proved
\[ = \frac{\sec^{2}{\left(2 x \right)} \frac{d}{d x} 2 x}{2} \]
chainApply the chain rule to the tangent function.✓ Proved
\[ = \sec^{2}{\left(2 x \right)} \]
derivative simplifyDifferentiate the inner function 2*x. Simplify the final expression.✓ Proved

Answer \( \tan^{2}{\left(2 x \right)} + 1 \)

Top: the graph of f = tan(2x)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = tan(2x)² + 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. A reviewer's objection to the explanation stands; see the receipt. 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
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 sec has poles at odd multiples of pi/2
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2
6	✓ 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: pass
qwen3.6:27b-mlx: fail (error) — Step 3 rewrites tan(2*x) as exp(log(tan(2*x))), which is an invalid algebraic identity for negative values of tan(2*x) and introduces unnecessary complexity. Furthermore, Step 4 applies the chain rule to the expression in Step 3 but produces the derivative of tan(2*x) directly, skipping the differentiation of the exponential and logarithmic layers, making the step logically inconsistent with the previous line.

Senior review claude-sonnet-5-5, 2026-10-11: fail — Every line is mathematically equivalent and the answer is correct. Step 3 is a pointless exp(log()) rewrite with a stray note about a 'previous error', and step 4 then discards that rewrite while applying the tan and chain rules together, so it mislabels what it does. The solution should have gone straight from line 2 to the chain rule on tan(2x).

qwen3.6:27b-mlx: uphold — The exp(log(tan(2x))) identity itself holds and domain concerns are not defects, but step 4 jumps from the differentiated exp(log(tan(2x))) to sec(2x)^2 * D(2x). That silently undoes the rewrite and applies the tan derivative along with the chain rule, so it does more than one change and is not a legitimate single 'chain' step.

Every verdict on record (7)

claude-sonnet-5-5: fail (misleading) 2026-10-11 — Every line is mathematically equivalent and the answer is correct. Step 3 is a pointless exp(log()) rewrite with a stray note about a 'previous error', and step 4 then discards that rewrite while applying the tan and chain rules together, so it mislabels what it does. The solution should have gone straight from line 2 to the chain rule on tan(2x).
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 rewrites tan(2*x) as exp(log(tan(2*x))), which is an invalid algebraic identity for negative values of tan(2*x) and introduces unnecessary complexity. Furthermore, Step 4 applies the chain rule to the expression in Step 3 but produces the derivative of tan(2*x) directly, skipping the differentiation of the exponential and logarithmic layers, making the step logically inconsistent with the previous line.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is mathematically invalid; it rewrites tan(2x) as exp(log(tan(2x))), which is not an identity (it fails for negative arguments and introduces complex numbers), and the note claims to fix a non-existent error. Furthermore, Step 4 applies the chain rule to the expression in Step 3 but produces the derivative of tan, not the derivative of exp(log(...)), making the step logically incoherent.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 rewrites tan(2*x) as exp(log(tan(2*x))), which is an invalid identity for negative values of tan(2*x) and introduces unnecessary complexity. Furthermore, Step 4 jumps from this rewritten form directly to the derivative of tan without showing the intermediate differentiation of the exponential/logarithmic composition, violating the one-rule-per-step constraint and logical flow.
gpt-oss:20b: fail (error) 2026-09-21 — Step 3 rewrites tan(2*x) as exp(log(tan(2*x))), but step 4 then applies the chain rule directly to tan(2*x) instead of to exp(log(tan(2*x))). This applies two separate transformations in one step and ignores the rewritten form, making the sequence logically inconsistent.

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.