2026 USAAIO Round 1 Sample problems, Problem 6

AI Olympiads
1

Problem 6

Compute the following derivatives.

Part 6.1

\frac{d \tanh x}{dx}.

Part 6.2

\nabla_{\theta}\sum_{n=0}^{N-1}(y^{(n)}-\theta^{\top}x^{(n)})^{2},

where \theta, x^{(n)}\in\mathbb{R}^{d} and y^{(n)}\in\mathbb{R}.

Reasoning is not required.

2

Part 6.1

\text{sech}^2(x)

Part 6.2

\nabla_{\theta} J(\theta) = 2 \sum_{n=0}^{N-1} (\theta^{\top}x^{(n)} - y^{(n)})x^{(n)}