The derivative from the limit definition
Problem 2.1635 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = \sqrt{x + 5} \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ \sqrt{h + x + 5} \]Write out f(x + h).✓ Proved
- \[ \frac{- \sqrt{x + 5} + \sqrt{h + x + 5}}{h} = \frac{1}{\sqrt{x + 5} + \sqrt{h + x + 5}} \]Rationalise the numerator and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} \frac{1}{\sqrt{x + 5} + \sqrt{h + x + 5}} = \frac{1}{2 \sqrt{x + 5}} \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = \frac{1}{2 \sqrt{x + 5}} \)
✓ Nihil obstat Lines: 3 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 |
| 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.
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.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 standard algebraic technique of rationalizing the numerator to resolve the indeterminate form. The steps are logically sound and lead to the correct result.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.