The derivative from the limit definition
Problem 2.1634 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = \sqrt{x + 6} \), then find \( \displaystyle f'(-1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ \sqrt{h + x + 6} \]Write out f(x + h).✓ Proved
- \[ \frac{- \sqrt{x + 6} + \sqrt{h + x + 6}}{h} = \frac{1}{\sqrt{x + 6} + \sqrt{h + x + 6}} \]Rationalise the numerator and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} \frac{1}{\sqrt{x + 6} + \sqrt{h + x + 6}} = \frac{1}{2 \sqrt{x + 6}} \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. \frac{1}{2 \sqrt{x + 6}} \right|_{\substack{ x=-1 }} = \frac{\sqrt{5}}{10} \]At x = -1.✓ Proved
Answer \( f'(x) = \frac{1}{2 \sqrt{x + 6}},\ f'(-1) = \frac{\sqrt{5}}{10} \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | matches the derivative by the power, quotient and chain rules |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the limit definition of the derivative, including the necessary algebraic step of rationalizing the numerator to resolve the 0/0 indeterminate form. The final evaluation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the limit definition of the derivative, including the necessary algebraic step of rationalizing the numerator to resolve the 0/0 indeterminate form. The final evaluation is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the limit definition of the derivative, including the algebraic step of rationalizing the numerator to resolve the 0/0 indeterminate form, and correctly evaluates the result at x = -1.gpt-oss:20b: pass 2026-10-05
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
generator:structured/derivative_definition, checked 2026-10-05 with SymPy 1.14.0.