Derivative of \( \displaystyle \left(4 x - 2\right)^{4 x - 3} \)
Problem 2.420 · medium
Differentiate \( \displaystyle f(x) = \left(4 x - 2\right)^{4 x - 3} \).
- \[ \frac{d}{d x} \left(4 x - 2\right)^{4 x - 3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]rewriteRewrite the power function using the exponential identity.≈ Checked numerically
- \[ = e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \frac{d}{d x} \left(4 x - 3\right) \ln{\left(4 x - 2 \right)} \]chainApply the chain rule to the exponential function.✓ Proved
- \[ = \left(\left(4 x - 3\right) \frac{d}{d x} \ln{\left(4 x - 2 \right)} + \ln{\left(4 x - 2 \right)} \frac{d}{d x} \left(4 x - 3\right)\right) e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\frac{4 \left(4 x - 3\right)}{4 x - 2} + 4 \ln{\left(4 x - 2 \right)}\right) e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]derivativeDifferentiate the individual terms in the product.✓ Proved
- \[ = \left(4 \ln{\left(4 x - 2 \right)} + \frac{16 x - 12}{4 x - 2}\right) e^{\left(4 x - 3\right) \ln{\left(4 x - 2 \right)}} \]algebraSimplify the product in the second term.✓ Proved
- \[ = \left(4 x - 2\right)^{4 x - 3} \left(4 \ln{\left(4 x - 2 \right)} + \frac{16 x - 12}{4 x - 2}\right) \]algebra simplify simplify algebraConvert the exponential term back to power form. Factor out the common 4 in the second term. Ensure the fraction is expanded for clarity. Simplify the denominator of the fraction.≈ Checked numerically
- \[ = \left(4 x - 2\right)^{4 x - 3} \left(4 \ln{\left(4 x - 2 \right)} + \frac{8 x - 6}{2 x - 1}\right) \]simplifyCancel the common factor of 2.≈ Checked numerically
Answer \( \left(4 x - 2\right)^{4 x - 3} \left(4 \log{\left(4 x - 2 \right)} + \frac{16 x - 12}{4 x - 2}\right) \)
Lines: 8 proved, 4 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*((4*x - 2)**(4*x - 3)*(4*x + 2*(2*x - 1)*log(4*x - 2) - 3) + (-4*x - 2*(2*x - 1)*log(4*x - 2) + 3)*exp((4*x - 3)*log(4*x - 2)))/(2*x - 1); 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 4*x - 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*((4*x - 2)**(4*x - 3)*(-4*x - 2*(2*x - 1)*log(4*x - 2) + 3) + (4*x + 2*(2*x - 1)*log(4*x - 2) - 3)*exp((4*x - 3)*log(4*x - 2)))/(2*x - 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x - 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0 |
| 11 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (2*(2*x - 1)*(4*x - 2)**(4*x - 3)*(4*x + 2*(2*x - 1)*log(4*x - 2) - 3) + (4*x - 2)**(4*x - 2)*(-4*x - 2*(2*x - 1)*log(4*x - 2) + 3))/(2*x - 1)**2; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x - 2 = 0 undefined where 2*x - 1 = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left (-2*(2*x - 1)*(4*x - 2)**(4*x - 3) + (4*x - 2)**(4*x - 2))*(4*x + 2*(2*x - 1)*log(4*x - 2) - 3)/(2*x - 1)**2; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x - 2 = 0 |
| 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: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies logarithmic differentiation, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary. The final simplification steps are algebraically sound and clearly labeled.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 applies two derivative rules in one line (derivative of 4*x-3 and derivative of log(4*x-2)), violating the one‑rule‑per‑step rule. The label "derivative" is also misleading because it masks the two separate derivative applications.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, product rule, and algebraic simplifications. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-20
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.