Derivative of arctan(x)
Arctangent's derivative is a small algebraic miracle: an inverse trig function goes in, and a plain rational function with no trig at all comes out.
Step-by-step solution
Let y = arctan(x), so tan(y) = x.
sec²(y) · y′ = 1, hence y′ = 1/sec²(y).
The identity sec²(y) = 1 + tan²(y) = 1 + x² gives y′ = 1/(1 + x²).
Why it works
The disappearance of all trigonometry from the answer is what makes this derivative important rather than merely cute: it means the area under the innocent-looking curve 1/(1+x²) is measured by arctangent. Integrate it from −∞ to ∞ and you get exactly π — the reason this function (the "Cauchy distribution" shape) keeps appearing in probability and physics.
The formula also captures arctangent's shape precisely. The derivative peaks at 1 when x = 0 (the curve has slope 1 at the origin) and decays like 1/x² in both directions, which is why arctan flattens out toward its horizontal asymptotes at ±π/2 — always rising, never arriving.
Range and composite inputs
The principal arctangent is defined for all real x and takes values in (−π/2, π/2) radians. The denominator 1 + x² is always positive, so its derivative exists and is positive everywhere.
For arctan(g(x)), use g′(x)/(1 + g(x)²) wherever g is defined and differentiable. For arctan(2x), this gives 2/(1 + 4x²). At x = 0 its slope is 2, twice the slope of arctan(x) at the origin.
For arctan(x), the point at x = 1 is (1, π/4) and the slope is 1/2, so its tangent line is y = π/4 + (x − 1)/2. A slope of 1 corresponds to a 45-degree tangent only when the graph axes use equal scales.
Common mistakes
- Writing the derivative as −1/(1+x²) — that belongs to arccotangent. Arctan's derivative is positive everywhere, consistent with its strictly increasing graph.
- Confusing arctan(x) = tan⁻¹(x) with 1/tan(x) = cot(x). The superscript −1 means inverse function, not reciprocal.
- Forgetting the chain rule on arctan(inner): d/dx[arctan(3x)] = 3/(1 + 9x²).
Practice problems
Differentiate arctan(2x)
Answer: 2/(1 + 4x²).
Find the slope of arctan(x) at x = 1
Answer: 1/(1+1) = 1/2.
Differentiate x arctan(x)
Answer: arctan(x) + x/(1 + x²) — product rule.