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The derivative from the limit definition

Problem 2.2496 · medium Video

Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = \sqrt{x + 2} \), then find \( \displaystyle f'(1) \).

By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
Reviewed
\[ \left. \sqrt{x + 2} \right|_{\substack{ x=h + x }} = \sqrt{h + x + 2} \]
Write out f(x + h).✓ Proved
\[ \frac{- \sqrt{x + 2} + \sqrt{h + x + 2}}{h} = \frac{1}{\sqrt{x + 2} + \sqrt{h + x + 2}} \]
Rationalise the numerator and cancel the factor h.✓ Proved
\[ \lim_{h \to 0^+} \frac{1}{\sqrt{x + 2} + \sqrt{h + x + 2}} = \frac{1}{2 \sqrt{x + 2}} \]
Now h → 0 by direct substitution.✓ Proved
\[ \left. 1 \frac{1}{2 \sqrt{x + 2}} \right|_{\substack{ x=1 }} = \frac{\sqrt{3}}{6} \]
At x = 1.✓ Proved

Answer \( f'(x) = \frac{1}{2 \sqrt{x + 2}},\ f'(1) = \frac{\sqrt{3}}{6} \)

Top: the graph of f = sqrt(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 1/(2sqrt(x + 2)).

✓ 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. How we verify

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: pass
qwen3.6:27b-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 (6)

qwen3.6:27b-mlx: pass 2026-10-26 — 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 2026-10-26
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies the limit definition of the derivative, handles the algebraic simplification (rationalizing the numerator) accurately, and evaluates the limit and specific value correctly.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification (rationalizing the numerator) accurately, and evaluates the limit and specific value correctly.
gpt-oss:20b: pass 2026-10-25

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 2026-10-26 with SymPy 1.14.0.