Derivative of \( \displaystyle \left(4 x - 1\right)^{\frac{1}{4 x - 1}} \)
Problem 2.490 · hard
Differentiate \( \displaystyle f(x) = \left(4 x - 1\right)^{\frac{1}{4 x - 1}} \).
- \[ \frac{d}{d x} \left(4 x - 1\right)^{\frac{1}{4 x - 1}} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}} \]rewriteRewrite the base and exponent using the exponential and logarithm.≈ Checked numerically
- \[ = e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}} \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{4 x - 1} \]productApply the product rule to the exponent.✓ Proved
- \[ = \left(\ln{\left(4 x - 1 \right)} \frac{d}{d x} \frac{1}{4 x - 1} + \frac{\frac{d}{d x} \ln{\left(4 x - 1 \right)}}{4 x - 1}\right) e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}} \]productApply the product rule to the terms in the parenthesis.✓ Proved
- \[ = \left(\ln{\left(4 x - 1 \right)} \frac{d}{d x} \frac{1}{4 x - 1} + \frac{4}{\left(4 x - 1\right)^{2}}\right) e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}} \]chain algebraDifferentiate the logarithm using the chain rule. Simplify the term inside the parentheses.✓ Proved
- \[ = \left(- \frac{4 \ln{\left(4 x - 1 \right)}}{\left(4 x - 1\right)^{2}} + \frac{4}{\left(4 x - 1\right)^{2}}\right) e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}} \]chainDifferentiate the reciprocal term using the chain rule.✓ Proved
- \[ = \frac{4 \left(1 - \ln{\left(4 x - 1 \right)}\right) e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}}}{\left(4 x - 1\right)^{2}} \]algebraFactor out the common term 4/(4*x-1)**2.✓ Proved
- \[ = \frac{4 \left(1 - \ln{\left(4 x - 1 \right)}\right) \left(4 x - 1\right)^{\frac{1}{4 x - 1}}}{\left(4 x - 1\right)^{2}} \]simplifySubstitute the exponential form back to the original base.≈ Checked numerically
Answer \( 4 \left(1 - \log{\left(4 x - 1 \right)}\right) \left(4 x - 1\right)^{-2 + \frac{1}{4 x - 1}} \)
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 4*((1 - log(4*x - 1))*(4*x - 1)**(1/(4*x - 1)) + (log(4*x - 1) - 1)*exp(log(4*x - 1)/(4*x - 1)))/(4*x - 1)**2; numeric agreement only, at 24 of 24 sampled points undefined where 4*x - 1 = 0 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 undefined where 4*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 9 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 4*((1 - log(4*x - 1))*exp(log(4*x - 1)/(4*x - 1)) + (4*x - 1)**(1/(4*x - 1))*(log(4*x - 1) - 1))/(4*x - 1)**2; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x - 1 = 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 4*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 (error) — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(g) is exp(g)·g', not a product of two separate functions. This mislabeling misrepresents the rule used and violates the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly labels the application of the chain rule (differentiating exp(u)) as the product rule. The step differentiates the outer exponential function, which requires the chain rule, not the product rule.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 incorrectly labels the application of the chain rule (differentiating exp(u)) as the product rule. The step differentiates the outer exponential function, which requires the chain rule, not the product rule.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) is exp(g)·g', not a product of two separate functions. This mislabeling misrepresents the rule used and violates the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule (derivative of exp(u)) as the 'product' rule. The step differentiates the outer exponential function, which is a chain rule application, not a product rule application.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the chain rule as a product rule, and step 9 incorrectly labels the substitution of the exponential back to the original base as a simplify step; both should be labeled "chain" and "rewrite", respectively.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule (differentiating exp(g(x))) as the 'product' rule. The product rule is applied in Step 4 to the exponent, but Step 3 is the outer derivative of the exponential function.gpt-oss:20b: fail (style) 2026-09-20 — Step 3 incorrectly labels the application of the chain rule as a product rule. The derivative of exp(u) is exp(u)*u', which is a single chain rule application, not a product rule. This mislabeling could mislead a student about which rule to apply.
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.