Derivative of \( \displaystyle \frac{2 \operatorname{asin}{\left(5 x \right)}}{5} \)
Problem 2.512 · medium
Differentiate \( \displaystyle f(x) = \frac{2 \operatorname{asin}{\left(5 x \right)}}{5} \).
- \[ \frac{d}{d x} \frac{2 \operatorname{asin}{\left(5 x \right)}}{5} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{2 \frac{d}{d x} \operatorname{asin}{\left(5 x \right)}}{5} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{2 \frac{d}{d x} 5 x}{5 \sqrt{1 - 25 x^{2}}} \]chainApply the chain rule to the inverse sine function.✓ Proved
- \[ = \frac{2}{\sqrt{1 - 25 x^{2}}} \]derivative algebra simplifyDifferentiate the inner function 5*x. Simplify the expression by canceling the 5s. Final simplified form.✓ Proved
Answer \( \frac{2}{\sqrt{1 - 25 x^{2}}} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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] undefined where 1 - 25*x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 25*x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 25*x**2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 25*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 25*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 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The algebraic simplifications are valid and properly labeled.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and basic derivative rules in separate steps. The algebraic simplifications are valid and properly labeled.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in distinct steps. The algebraic simplifications are handled separately and correctly labeled.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-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.