Derivative of \( \displaystyle \sqrt{1 - \left(4 x + 1\right)^{2}} + \left(4 x + 1\right) \operatorname{asin}{\left(4 x + 1 \right)} \)
Problem 2.434 · hard
Differentiate \( \displaystyle f(x) = \sqrt{1 - \left(4 x + 1\right)^{2}} + \left(4 x + 1\right) \operatorname{asin}{\left(4 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(\sqrt{1 - \left(4 x + 1\right)^{2}} + \left(4 x + 1\right) \operatorname{asin}{\left(4 x + 1 \right)}\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} \left(4 x + 1\right) \operatorname{asin}{\left(4 x + 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 1\right)^{2}} \]sumApply the sum rule.✓ Proved
- \[ = \left(4 x + 1\right) \frac{d}{d x} \operatorname{asin}{\left(4 x + 1 \right)} + \operatorname{asin}{\left(4 x + 1 \right)} \frac{d}{d x} \left(4 x + 1\right) + \frac{d}{d x} \sqrt{1 - \left(4 x + 1\right)^{2}} \]productApply the product rule to the second term.✓ Proved
- \[ = \left(4 x + 1\right) \frac{d}{d x} \operatorname{asin}{\left(4 x + 1 \right)} + 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 1\right)^{2}} \]constant rewriteDifferentiate the linear term 4x + 1. Rewrite the argument of asin using exp and log for consistency.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 1\right)^{2}} + \frac{\left(4 x + 1\right) \frac{d}{d x} \left(4 x + 1\right)}{\sqrt{1 - \left(4 x + 1\right)^{2}}} \]chainApply the chain rule to the inverse sine term.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 1\right)^{2}} + \frac{4 \left(4 x + 1\right)}{\sqrt{1 - \left(4 x + 1\right)^{2}}} \]constantDifferentiate the inner function 4x + 1.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{d}{d x} \sqrt{1 - \left(4 x + 1\right)^{2}} + \frac{16 x + 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} \]rewrite rewriteRewrite the square root as a power. Rewrite the denominator using a negative exponent.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{16 x + 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} + \frac{\frac{d}{d x} \left(1 - \left(4 x + 1\right)^{2}\right)}{2 \sqrt{1 - \left(4 x + 1\right)^{2}}} \]chainApply the chain rule to the power term.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{16 x + 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} + \frac{\frac{d}{d x} 1}{2 \sqrt{1 - \left(4 x + 1\right)^{2}}} - \frac{\frac{d}{d x} \left(4 x + 1\right)^{2}}{2 \sqrt{1 - \left(4 x + 1\right)^{2}}} \]sumApply the sum rule to the derivative of the radicand.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{16 x + 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} - \frac{\frac{d}{d x} \left(4 x + 1\right)^{2}}{2 \sqrt{1 - \left(4 x + 1\right)^{2}}} \]constant algebraThe derivative of the constant 1 is 0. Simplify the expression by removing the zero term.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{16 x + 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} - \frac{32 x + 8}{2 \sqrt{1 - \left(4 x + 1\right)^{2}}} \]chain constantApply the chain rule to the squared term. Multiply the constants in the derivative term.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} + \frac{- 16 x - 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} + \frac{16 x + 4}{\sqrt{1 - \left(4 x + 1\right)^{2}}} \]algebraSimplify the first term by multiplying 1/2 and 8.✓ Proved
- \[ = 4 \operatorname{asin}{\left(4 x + 1 \right)} \]simplifyCombine the two identical terms that cancel each other out.✓ Proved
Answer \( 4 \operatorname{asin}{\left(4 x + 1 \right)} \)
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. 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 | ✓ 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] |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - (4*x + 1)**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 asin is real only on [-1, 1] |
| 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 5 rewrites asin(4*x+1) as asin(exp(log(4*x+1))), altering the function. The subsequent chain rule application in step 6 then uses the derivative of asin(exp(log(4*x+1))) instead of asin(4*x+1), producing an incorrect derivative. This is a multi‑rule error and invalidates the solution.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] Step 5 is mathematically invalid because `exp(log(u))` is not equal to `u` for negative `u`, and the domain of `asin` includes negative arguments. This rewrite changes the function being differentiated.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — [domain objection, downgraded to style] Step 5 is mathematically invalid because `exp(log(u))` is not equal to `u` for negative `u`, and the domain of `asin` includes negative arguments. This rewrite changes the function being differentiated.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 rewrites asin(4*x+1) as asin(exp(log(4*x+1))), altering the function. The subsequent chain rule application in step 6 then uses the derivative of asin(exp(log(4*x+1))) instead of asin(4*x+1), producing an incorrect derivative. This is a multi‑rule error and invalidates the solution.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the final simplification is correct.gpt-oss:20b: fail (style) 2026-09-20 — [domain objection, downgraded to style] Step 5 rewrites asin(4*x+1) as asin(exp(log(4*x+1))) and then immediately applies the chain rule. This combines a rewrite and a differentiation in a single step, violating the rule that each step must change only one thing. The rewrite also introduces an extra domain restriction that is not justified here.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 4 is labeled 'constant' but applies the power rule to differentiate a linear term; 'power' or 'derivative' would be more accurate. Step 5 introduces an unnecessary and confusing rewrite of asin using exp/log, which is not a standard differentiation step and obscures the logic.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 rewrites the argument of asin from 4*x+1 to exp(log(4*x+1)), changing the function. This is not a valid rewrite and propagates an incorrect expression through the rest of the derivation.
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.