Derivative of \( \displaystyle \operatorname{atan}{\left(5 x - 3 \right)} \)
Problem 2.855 · medium Video
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(5 x - 3 \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}{\left(5 x - 3 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(5 x - 3\right)}{\left(5 x - 3\right)^{2} + 1} \]chainApply the chain rule for the arctangent function.✓ Proved
- \[ = \frac{5}{\left(5 x - 3\right)^{2} + 1} \]derivative algebraDifferentiate the inner function 5*x - 3. Multiply the terms together.✓ Proved
- \[ = \frac{5}{25 x^{2} - 30 x + 10} \]algebra simplifyExpand the denominator. Combine the constant terms.✓ Proved
Answer \( \frac{5}{\left(5 x - 3\right)^{2} + 1} \)
✓ 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. 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 3)**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 3)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 3)**2 + 1 = 0 undefined where 25*x**2 - 30*x + 10 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 25*x**2 - 30*x + 10 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (5*x - 3)**2 + 1 = 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 chain rule and simplifies the expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule and simplifies the expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26
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 2026-09-27 with SymPy 1.14.0.