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The derivative from the limit definition

Problem 2.2489 · medium Video

Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = \frac{4}{x - 1} \).

By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
Reviewed
\[ \left. \frac{4}{x - 1} \right|_{\substack{ x=h + x }} = \frac{4}{h + x - 1} \]
Write out f(x + h).✓ Proved
\[ \frac{\frac{4}{h + x - 1} - \frac{4}{x - 1}}{h} = - \frac{4}{\left(x - 1\right) \left(h + x - 1\right)} \]
Combine and cancel the factor h.✓ Proved
\[ \lim_{h \to 0^+}\left(- \frac{4}{\left(x - 1\right) \left(h + x - 1\right)}\right) = - \frac{4}{\left(x - 1\right)^{2}} \]
Now h → 0 by direct substitution.✓ Proved

Answer \( f'(x) = - \frac{4}{\left(x - 1\right)^{2}} \)

Top: the graph of f = 4/(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -4/(x - 1)².

✓ 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. 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
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, performs the algebraic simplification accurately, and evaluates the limit correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the limit correctly.
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, performs the algebraic simplification accurately, and evaluates the limit properly.
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 accurately, and evaluates the limit 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.