Home›Calculus 1›Power rule›Problem 2.295

Derivative of \( \displaystyle \frac{5 \sqrt{4 x^{2} + 1}}{2} \)

Problem 2.295 · medium Video

Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{4 x^{2} + 1}}{2} \).

\[ \frac{d}{d x} \frac{5 \sqrt{4 x^{2} + 1}}{2} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \sqrt{4 x^{2} + 1}}{2} \]
constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(4 x^{2} + 1\right)}{4 \sqrt{4 x^{2} + 1}} \]
chain algebraApply the chain rule. Multiply the constants.✓ Proved
\[ = \frac{10 x}{\sqrt{4 x^{2} + 1}} \]
power algebra simplify rewriteDifferentiate the inner function. Rearrange the terms. Simplify the expression. Convert the negative exponent back to a square root in the denominator.✓ Proved

Answer \( \frac{10 x}{\sqrt{4 x^{2} + 1}} \)

Top: the graph of f = 5sqrt(4x² + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 10x/sqrt(4x² + 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
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4x*2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4x*2 + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4x*2 + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4x*2 + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4x*2 + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 4x*2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 4x*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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-06: pass — The derivation is correct: constant-multiple, rewrite to a fractional power, chain rule, then evaluate the inner derivative 8x and tidy up. All lines were proved equivalent, and the labels are acceptable.

unspecified panel model: dismiss — No concrete defect was stated, and SymPy proved every line and the final answer equivalent. The labels (constant-multiple, rewrite, chain, algebra, power, simplify) each name a rule that was actually applied, and no step does two independent things.

Every verdict on record (13)

claude-sonnet-5-5: pass 2026-10-06 — The derivation is correct: constant-multiple, rewrite to a fractional power, chain rule, then evaluate the inner derivative 8x and tidy up. All lines were proved equivalent, and the labels are acceptable.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label.
gpt-oss:20b: fail (error) 2026-09-19 — Step 6 applies two differentiation rules (constant‑multiple and power) at once and labels the step as "power", which is not a valid single rule from the allowed vocabulary.
gpt-oss:20b: pass 2026-09-17
deepseek-r1:70b: pass 2026-09-17

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 2026-09-26 with SymPy 1.14.0.