BPTT — the tanh backward factor

Background

Every BPTT step multiplies by the local derivative of the cell's nonlinearity. For tanh, the derivative is expressible from the activation $h = \tanh(z)$ itself:

$$\frac{\partial h}{\partial z} = 1 - \tanh(z)^2 = 1 - h^2$$

(In the lesson's training loop this is the dzh = dh * (1 - h**2) line.) Because $1 - h^2 \le 1$ always, this factor is one source of vanishing gradients.

Problem statement

Implement tanh_grad(h) returning $1 - h^2$ elementwise, where h is the tanh activation.

Input

• h — array-like (or scalar) of tanh activations, each in $(-1, 1)$.

Output

Returns an np.ndarray (same shape) of local gradients $1 - h^2$.

Examples

Example 1 — steepest at 0

Input:  h = 0
Output: 1.0

Example 2

Input:  h = [0, 0.5, -0.5]
Output: [1.   0.75 0.75]

Constraints

• Use the cached form 1 - h**2 — no need to recompute tanh.
• Return an np.ndarray.

Notes

• The derivative peaks at 1 (when $h = 0$) and collapses toward 0 as the unit saturates ($|h|\to1$) — exactly where gradient flow dies.