Home›Calculus 1›Chain rule›Problem 2.2264

Derivative of \( \displaystyle \frac{5 \sqrt{16 x^{2} + 8 x + 2}}{4} \)

Problem 2.2264 · hard Video

Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{16 x^{2} + 8 x + 2}}{4} \).

\[ \frac{d}{d x} \frac{5 \sqrt{16 x^{2} + 8 x + 2}}{4} \]
Start with the derivative of the function.✓ Proved
\[ = \frac{5 \frac{d}{d x} \sqrt{16 x^{2} + 8 x + 2}}{4} \]
constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(16 x^{2} + 8 x + 2\right)}{8 \sqrt{16 x^{2} + 8 x + 2}} \]
chain algebraApply the chain rule. Multiply the constants.✓ Proved
\[ = \frac{5 \left(32 x + 8\right)}{8 \sqrt{16 x^{2} + 8 x + 2}} \]
derivativeDifferentiate the polynomial inside.✓ Proved
\[ = \frac{5 \left(4 x + 1\right)}{\sqrt{16 x^{2} + 8 x + 2}} \]
algebra algebraFactor out 8 from the polynomial. Simplify the constants.✓ Proved
\[ = \frac{20 x + 5}{\sqrt{16 x^{2} + 8 x + 2}} \]
simplifyRewrite with a positive exponent in the denominator.✓ Proved

Answer \( \frac{20 x + 5}{\sqrt{16 x^{2} + 8 x + 2}} \)

Top: the graph of f = 5sqrt(16x² + 8x + 2)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5(16x + 4)/(4sqrt(16x² + 8x + 2)).

✓ 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 16x2 + 8x + 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x2 + 8x + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x2 + 8x + 2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x2 + 8x + 2 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x2 + 8x + 2 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x2 + 8x + 2 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 16x2 + 8x + 2 = 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.