Home›Calculus 1›Quotient rule›Problem 2.568

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

Problem 2.568 · hard Video

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

\[ \frac{d}{d x} \frac{\cos{\left(2 x + 2 \right)}}{2 x + 2} \]
derivative quotientStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
\[ = \cos{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{2 x + 2} + \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} \]
productApply the product rule.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} - \frac{\cos{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{\left(2 x + 2\right)^{2}} \]
chainApply the chain rule to the second term.✓ Proved
\[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 x + 2} - \frac{2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
derivativeDifferentiate the inner function 2*x + 2.✓ Proved
\[ = - \frac{2 \sin{\left(2 x + 2 \right)}}{2 x + 2} - \frac{2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
chain algebra algebraApply the chain rule to the first term. Distribute and simplify the terms. Rewrite negative exponents as denominators.✓ Proved
\[ = \frac{- 2 \left(2 x + 2\right) \sin{\left(2 x + 2 \right)} - 2 \cos{\left(2 x + 2 \right)}}{\left(2 x + 2\right)^{2}} \]
algebra simplifyFind a common denominator. Factor out the common term -2.✓ Proved

Answer \( \frac{\left(- 2 x - 2\right) \sin{\left(2 x + 2 \right)} - \cos{\left(2 x + 2 \right)}}{2 \left(x + 1\right)^{2}} \)

Top: the graph of f = cos(2x + 2)/(2x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -2sin(2x + 2)/(2x + 2) - 2cos(2x + 2)/(2x + 2)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. 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 undefined where 2*x + 2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 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: fail (style) — Step 2 incorrectly labels the rewrite of the quotient as a "quotient" rule; it should be labeled "rewrite". No other mathematical errors are present.
qwen3.6:27b-mlx: fail (style) — Step 2 is labeled 'quotient' but performs an algebraic rewrite of a fraction into a product with a negative exponent; the label should be 'rewrite' or 'algebra'. Step 4 is labeled 'chain' but applies the power rule to the outer function and the chain rule to the inner function simultaneously; this violates the one-rule-per-step constraint and should be split or labeled 'power' (with chain applied in a subsequent step).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 2 is labeled 'quotient' but performs an algebraic rewrite of a fraction into a product with a negative exponent; the label should be 'rewrite' or 'algebra'. Step 4 is labeled 'chain' but applies the power rule to the outer function and the chain rule to the inner function simultaneously; this violates the one-rule-per-step constraint and should be split or labeled 'power' (with chain applied in a subsequent step).
gpt-oss:20b: fail (style) 2026-09-26 — Step 2 incorrectly labels the rewrite of the quotient as a "quotient" rule; it should be labeled "rewrite". No other mathematical errors are present.
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 2 is labeled 'quotient' but performs an algebraic rewrite (converting a fraction to a negative exponent); the label should be 'rewrite' or 'algebra'. Step 4 is labeled 'chain' but applies the power rule to the outer function; the label should be 'power'.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21

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.