Derivative of \( \displaystyle \frac{\operatorname{asin}{\left(4 x \right)}}{4} \)
Problem 2.557 · medium
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{4} \).
- \[ \frac{d}{d x} \frac{\operatorname{asin}{\left(4 x \right)}}{4} \]constant-multiplePull out the constant factor 1/4.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(4 x \right)}}{4} \]chain rewriteApply the chain rule to the inner function 4*x. Not applicable here, but let's differentiate the composition directly.✓ Proved
- \[ = \frac{\frac{d}{d x} 4 x}{4 \sqrt{1 - 16 x^{2}}} \]chainApply the derivative formula for arcsin(u) where u = 4x.✓ Proved
- \[ = \frac{1}{\sqrt{1 - 16 x^{2}}} \]derivative simplifyDifferentiate the inner function 4*x. Simplify the expression by canceling the 4s.✓ Proved
Answer \( \frac{1}{\sqrt{1 - 16 x^{2}}} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - 16*x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 16*x**2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 16*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 16*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: fail (error) — Step 3 rewrites asin(4*x) as exp(log(asin(4*x))), which is mathematically incorrect and applies more than one rule at once. This step should be removed or replaced with the correct chain rule application.qwen3.6:27b-mlx: fail (error) — Step 3 introduces an unnecessary and incorrect rewrite of asin(4*x) as exp(log(asin(4*x))), which is not a standard differentiation step and serves no purpose in the chain rule application. This step is logically flawed and misleading.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 introduces an unnecessary and incorrect rewrite of asin(4*x) as exp(log(asin(4*x))), which is not a standard differentiation step and serves no purpose in the chain rule application. This step is logically flawed and misleading.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 rewrites asin(4*x) as exp(log(asin(4*x))), which is mathematically incorrect and applies more than one rule at once. This step should be removed or replaced with the correct chain rule application.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 introduces an unnecessary and incorrect rewrite of arcsin as exp(log(asin(...))), which is not a valid differentiation step and breaks the logical flow. Step 2 is also defective because it labels the step 'chain' but does not actually apply the chain rule; it merely sets up the derivative, making the label incorrect for the transformation shown.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two transformations at once: it rewrites asin(4*x) as exp(log(asin(4*x))) and then immediately differentiates. This violates the rule that each step must change only one thing. Additionally, the label "rewrite" is not appropriate for the subsequent differentiation that occurs in the same step.
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.