Machine Learning Basics
When the derivative dy/dx of y = (2x + 1)² at x = 1 is found by the chain rule with u = 2x + 1 and y = u², which value is correct?
- A6
- B9
- C12
Show answer and explanation
Answer: C. 12
By the chain rule, dy/dx = (dy/du) × (du/dx). dy/du = 2u and du/dx = 2, and at x = 1, u = 3, so dy/dx = 2 × 3 × 2 = 12. Backpropagation applies this chain rule step by step from the output side toward the input side to compute efficiently the gradient of the loss with respect to the weights of every layer. Each layer multiplies the gradient received from the layer after it by its own local derivative and passes the result to the layer before it.
- A. This finds only dy/du = 2u = 6 and forgets to multiply by du/dx = 2.
- B. This is the value of y at x = 1, 3² = 9, not the derivative.
- C. Correct. 2u × 2 = 2 × 3 × 2 = 12.