Derivative of \( \displaystyle \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \)
Problem 2.562 · hard
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 numerically
- \[ = e^{\ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)}} \frac{d}{d x} \ln{\left(2 x - 3 \right)} \cos{\left(2 x - 3 \right)} \]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 numerically
Answer \( \left(2 x - 3\right)^{\cos{\left(2 x - 3 \right)}} \left(- 2 \log{\left(2 x - 3 \right)} \sin{\left(2 x - 3 \right)} + \frac{2 \cos{\left(2 x - 3 \right)}}{2 x - 3}\right) \)
Lines: 8 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*((2*x - 3)**cos(2*x - 3)*(-(2*x - 3)*log(2*x - 3)*sin(2*x - 3) + cos(2*x - 3)) + ((2*x - 3)*log(2*x - 3)*sin(2*x - 3) - cos(2*x - 3))*exp(log(2*x - 3)*cos(2*x - 3)))/(2*x - 3); numeric agreement only, at 24 of 24 sampled points 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 numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*((2*x - 3)**cos(2*x - 3)*((2*x - 3)*log(2*x - 3)*sin(2*x - 3) - cos(2*x - 3)) + (-(2*x - 3)*log(2*x - 3)*sin(2*x - 3) + cos(2*x - 3))*exp(log(2*x - 3)*cos(2*x - 3)))/(2*x - 3); numeric agreement only, at 24 of 24 sampled points 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 (error) — 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 (style) — 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'.
Every verdict on record (4)
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.