Home›Calculus 1›Quotient rule›Problem 2.1088

Derivative of \( \displaystyle 3 \operatorname{atan}{\left(x + 2 \right)} \)

Problem 2.1088 · medium Video

Differentiate \( \displaystyle f(x) = 3 \operatorname{atan}{\left(x + 2 \right)} \).

\[ \frac{d}{d x} 3 \operatorname{atan}{\left(x + 2 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 3 \frac{d}{d x} \operatorname{atan}{\left(x + 2 \right)} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{3 \frac{d}{d x} \left(x + 2\right)}{\left(x + 2\right)^{2} + 1} \]
chainApply the chain rule for the arctangent function.✓ Proved
\[ = \frac{3}{\left(x + 2\right)^{2} + 1} \]
derivative algebraDifferentiate the inner function x + 2. Simplify the expression by multiplying.✓ Proved
\[ = \frac{3}{x^{2} + 4 x + 5} \]
algebra simplifyExpand the squared binomial. Combine the constant terms.✓ Proved

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

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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the constant multiple rule, chain rule, and basic differentiation rules in separate steps. All labels are valid and the algebraic simplifications are correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-28

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 2026-09-29 with SymPy 1.14.0.