Home›Calculus 1›Logarithmic differentiation›Problem 2.562

Derivative of \( \displaystyle \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \)

Problem 2.562 · hard Video

Differentiate \( \displaystyle f(x) = \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \).

\[ \frac{d}{d x} \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \]
rewriteRewrite the function using the exponential identity.✓ Proved
\[ = \frac{d}{d x} e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \]
derivativeApply the derivative to the exponential function.✓ Checked independently
\[ = e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \frac{d}{d x} \ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)} \]
productApply the product rule to the exponent.✓ Proved
\[ = \left(\ln{\left(2 x - 3 \right)} \frac{d}{d x} \cos{\left(2 x - 3 \right)} + \cos{\left(2 x - 3 \right)} \frac{d}{d x} \ln{\left(2 x - 3 \right)}\right) e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \]
sumApply the sum rule to the product of derivatives.✓ Proved
\[ = \left(\ln{\left(2 x - 3 \right)} \frac{d}{d x} \cos{\left(2 x - 3 \right)} + \frac{\cos{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 x - 3}\right) e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \left(\ln{\left(2 x - 3 \right)} \frac{d}{d x} \cos{\left(2 x - 3 \right)} + \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3}\right) e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \]
derivativeDifferentiate the inner linear function.✓ Proved
\[ = \left(- 2 \ln{\left(2 x - 3 \right)} \sin{\left(2 x - 3 \right)} + \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3}\right) e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \]
trig algebraDifferentiate the cosine function. Simplify the expression algebraically.✓ Proved
\[ = \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \left(- 2 \ln{\left(2 x - 3 \right)} \sin{\left(2 x - 3 \right)} + \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3}\right) \]
simplifyConvert the exponential back to the original power form.✓ Checked independently

Answer \( \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \left(- 2 \log{\left(2 x - 3 \right)} \sin{\left(2 x - 3 \right)} + \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3}\right) \)

Top: the graph of f = (2x - 3)^cos(2x - 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (2x - 3)^cos(2x - 3)·(-2log(2x - 3)·sin(2x - 3) + 2cos(2x - 3)/(2x - 3)).

Lines: 8 proved, 2 checked by independent solution. 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	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: simplify left 2((2x - 3)*cos(2x - 3)(-(2x - 3)log(2x - 3)sin(2x - 3) + cos(2x - 3)) + ((2x - 3)log(2x - 3)sin(2x - 3) - cos(2x - 3))exp(log(2x - 3)cos(2x - 3)))/(2x - 3); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: derivative of u*cos(u) matches derivative of exp(cos(u)log(u)) after exp(coslog u)=u*cos identity (definition of real power) (`checks/2.562/line2-7a30f991.py`) log is undefined for non-positive arguments
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 0
9	✓ Checked independently	claude-sonnet-5-5 script, run by sympy 1.14.0, sympy 1.14.0	sympy 1.14.0: simplify left 2((2x - 3)*cos(2x - 3)((2x - 3)log(2x - 3)sin(2x - 3) - cos(2x - 3)) + (-(2x - 3)log(2x - 3)sin(2x - 3) + cos(2x - 3))exp(log(2x - 3)cos(2x - 3)))/(2x - 3); numeric agreement only, at 24 of 24 sampled points claude-sonnet-5-5 script, run by sympy 1.14.0: line 8 with exp(cos(u)log(u)) replaced by ucos(u) (identity bc = exp(clog b)) equals line 9 (`checks/2.562/line9-a66a6911.py`) log is undefined for non-positive arguments undefined where 2*x - 3 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 3 = 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) — The solution mislabels several differentiation rules: step 3 applies the chain rule (derivative of exp), not the product rule; step 4 applies the product rule to the product cos(2*x‑3)*log(2*x‑3), not the sum rule. These labeling errors violate the contract’s requirement that each step name the rule actually applied.
qwen3.6:27b-mlx: fail (error) — Step 3 is labeled 'product', but the operation performed is the chain rule on the outer exponential function, not the product rule. The product rule is applied in Step 4.

Senior review claude-sonnet-5-5, 2026-10-10: fail — Every line is mathematically correct and the final answer matches. However, step 3 is labeled 'product' when it is the exponential/chain rule, and step 4 is labeled 'sum' when it is the product rule, so a student would learn the wrong rule names.

gpt-oss:20b: uphold — Both claims hold: step 3 differentiates exp(g) to exp(g)*g', which is the exponential/chain rule and not the product rule, and step 4 splits the derivative of cos*log into f'g+fg', which is the product rule and not the sum rule.
qwen3.6:27b-mlx: uphold — Step 3 is labeled 'product' but only applies the derivative of exp to the exponent; the product rule is first used in step 4. The math is right, so 'error' overstates it.

Every verdict on record (7)

claude-sonnet-5-5: fail (misleading) 2026-10-10 — Every line is mathematically correct and the final answer matches. However, step 3 is labeled 'product' when it is the exponential/chain rule, and step 4 is labeled 'sum' when it is the product rule, so a student would learn the wrong rule names.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 is labeled 'product', but the operation performed is the chain rule on the outer exponential function, not the product rule. The product rule is applied in Step 4.
gpt-oss:20b: fail (style) 2026-09-26 — The solution mislabels several differentiation rules: step 3 applies the chain rule (derivative of exp), not the product rule; step 4 applies the product rule to the product cos(2*x‑3)*log(2*x‑3), not the sum rule. These labeling errors violate the contract’s requirement that each step name the rule actually applied.
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 2 is labeled 'derivative' but performs no differentiation; it merely presents the expression to be differentiated, so it should be labeled 'rewrite'. Step 3 is labeled 'product' but applies the chain rule to the outer exponential function, so it should be labeled 'chain'.
gpt-oss:20b: fail (error) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(g(x)) is exp(g(x))*g'(x), which is a chain rule application, not a product rule. This mislabeling constitutes a defect in the solution.
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 is labeled 'derivative' but performs no differentiation; it merely rewrites the expression from step 1. Step 3 is labeled 'product' but applies the chain rule to the exponential function. The labels are swapped and incorrect for the operations performed.
gpt-oss:20b: fail (error) 2026-09-21 — Step 3 incorrectly labels the chain rule as a product rule, and step 4 incorrectly labels the product rule as a sum rule. Both steps apply a single rule but use the wrong name, violating the contract’s labeling requirements.

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.