Leibniz rule at a moving boundary
When an integral's boundary depends on x, its derivative measures the integrand at that boundary and multiplies it by the boundary's speed.
If F(x) = ∫₀ᵍ⁽ˣ⁾ f(t) dt, then F′(x) = f(g(x)) · g′(x).
For g(x) = sin(x) and f(t) = 1/(1+t²), we obtain F′(x) = cos(x)/(1 + sin²(x)). At x = π/6, the value is 2√3/5 ≈ 0.69282.
The fundamental theorem supplies the boundary value; the chain rule supplies its velocity.