Derivative of \( \displaystyle \left(4 x - 1\right)^{\frac{1}{4 x - 1}} \)
Problem 2.490 · hard Video
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 independently
- \[ = e^{\frac{\ln{\left(4 x - 1 \right)}}{4 x - 1}} \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{4 x - 1} \]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 independently
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 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 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 claude-sonnet-5-5 script, run by sympy 1.14.0: exp(log(b)/b) replaced by b**(1/b); functions and derivatives agree: True True ( checks/2.490/line2-8affb8f9.py)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 independently | claude-sonnet-5-5 script, run by sympy 1.14.0, 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 claude-sonnet-5-5 script, run by sympy 1.14.0: after rewriting exp(log(b)/b) as b**(1/b), difference simplifies to 0 ( checks/2.490/line9-19f8f478.py)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.
Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct and the final answer matches, but step 3 is labeled 'product' when it applies the chain/exponential rule, and its note ('apply the product rule to the exponent') misdescribes the step. A student would learn the wrong rule for differentiating exp(g).
gpt-oss:20b: uphold — Step 3 differentiates exp(g) into exp(g)*g', which is the chain/exponential rule; the product rule on g is only applied in step 4, so the 'product' label is wrong.qwen3.6:27b-mlx: uphold — Same defect: the step differentiates the outer exponential (chain rule) and leaves the inner Derivative of g untouched, so naming it 'product' misnames the rule.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-09 — The math is correct and the final answer matches, but step 3 is labeled 'product' when it applies the chain/exponential rule, and its note ('apply the product rule to the exponent') misdescribes the step. A student would learn the wrong rule for differentiating exp(g).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.