Home›Calculus 1›Chain rule›Problem 2.889

Derivative of \( \displaystyle \frac{\sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \)

Problem 2.889 · medium Video

Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \).

\[ \frac{d}{d x} \frac{\sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \]
constant-multiple rewritePull out the constant factor 1/2. Rewrite the square root as a fractional power.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\left(2 x - 3\right)^{2} + 1\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
chainApply the chain rule.✓ Proved
\[ = \frac{\frac{d}{d x} \left(2 x - 3\right)^{2}}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = \frac{\left(4 x - 6\right) \frac{d}{d x} \left(2 x - 3\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
powerApply the power rule to the squared term.✓ Proved
\[ = \frac{8 x - 12}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
\[ = \frac{2 x - 3}{\sqrt{\left(2 x - 3\right)^{2} + 1}} \]
simplifySimplify the expression to its final form.✓ Proved

Answer \( \frac{2 x - 3}{\sqrt{\left(2 x - 3\right)^{2} + 1}} \)

Top: the graph of f = sqrt((2x - 3)² + 1)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = (4x - 6)/(2sqrt((2x - 3)² + 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 (2x - 3)*2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (2x - 3)*2 + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (2x - 3)*2 + 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (2x - 3)*2 + 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (2x - 3)*2 + 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (2x - 3)*2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where (2x - 3)*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
qwen3.6:27b-mlx: pass — The solution correctly applies the rules in a granular manner, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.

Senior review claude-sonnet-5-5, 2026-10-13: pass — The derivative is correct: (1/4)·u^(-1/2)·4(2x-3) = (2x-3)/sqrt(u), with u = (2x-3)^2 + 1. Every step is a single change with an acceptable rule label, and SymPy proved every line. The panel listed no specific objection to rule on, and there are no weak lines.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — The derivative is correct: (1/4)·u^(-1/2)·4(2x-3) = (2x-3)/sqrt(u), with u = (2x-3)^2 + 1. Every step is a single change with an acceptable rule label, and SymPy proved every line. The panel listed no specific objection to rule on, and there are no weak lines.
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the rules in a granular manner, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies the sum rule but omits the derivative of the constant term 1. The derivative of 1 is 0, so the step should either include it explicitly or note that it vanishes. This omission makes the step mathematically incorrect.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 5 drops the derivative of the constant term +1. The chain rule produced a factor of ((2*x-3)**2+1)**(-1/2) times Derivative((2*x-3)**2+1,x), but the solution replaces it with Derivative((2*x-3)**2,x), effectively applying the sum rule incorrectly by omitting the derivative of the constant. This is a mathematical error.

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 2026-09-27 with SymPy 1.14.0.