Study Plan

Differentiate a Tanh Activation

Activation Functions

Easy

Evaluate scalar tanh and manually propagate upstream_gradient through the activation.

y = \tanh(x)

\frac{\partial L}{\partial x} = g(1-y^2)

Here x is the scalar input, y is its tanh output, L is the final loss, and g is upstream_gradient, the derivative of L with respect to y.

Return a tuple of two zero-dimensional PyTorch tensors: y first, then the gradient with respect to x. Both must preserve the inputs’ floating-point dtype and device.

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Examples

Input: x = 0.5, upstream_gradient = 2.0

Output: (0.462117, 1.572895)

Explanation: The upstream gradient scales the local tanh derivative by two.

Input: x = 0.0, upstream_gradient = -3.0

Output: (0.0, -3.0)

Input: x = 8.0, upstream_gradient = 1.25

Output: (1.0, 0.000001)

Hint 1

torch.tanh preserves a tensor’s shape and dtype.

Hint 2

Tensor.square is useful for the local derivative.

Requirements

Use PyTorch and compute the gradient manually, without automatic differentiation.
Preserve the supplied float16, float32, or float64 dtype; examples use float64.
Do not modify x or upstream_gradient.

Constraints

x and upstream_gradient are finite zero-dimensional tensors sharing a dtype and device.
The dtype is float16, float32, or float64.

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