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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Copyright 2018 The TensorFlow Authors.
Step1:  Automatic differentiation and gradient tape
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    <a target="_blank" href="https
Step2:  Gradient tapes
TensorFlow provides the tf.GradientTape API for automatic differentiation - computing the gradient of a computation with respect to its input variables. Tensorflow "records" all operations executed inside the context of a tf.GradientTape onto a "tape". Tensorflow then uses that tape and the gradients associated with each recorded operation to compute the gradients of a "recorded" computation using reverse mode differentiation.
For example
Step3:  You can also request gradients of the output with respect to intermediate values computed during a "recorded" tf.GradientTape context.
Step4:  By default, the resources held by a GradientTape are released as soon as GradientTape.gradient() method is called. To compute multiple gradients over the same computation, create a persistent gradient tape. This allows multiple calls to the gradient() method as resources are released when the tape object is garbage collected. For example
Step5:  Recording control flow
Because tapes record operations as they are executed, Python control flow (using ifs and whiles for example) is naturally handled
Step6:  Higher-order gradients
Operations inside of the GradientTape context manager are recorded for automatic differentiation. If gradients are computed in that context, then the gradient computation is recorded as well. As a result, the exact same API works for higher-order gradients as well. For example
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Python Code:
#@title Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
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# https://www.apache.org/licenses/LICENSE-2.0
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Explanation: Copyright 2018 The TensorFlow Authors.
End of explanation
import tensorflow.compat.v1 as tf
Explanation: Automatic differentiation and gradient tape
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    <a target="_blank" href="https://colab.research.google.com/github/tensorflow/docs/blob/master/site/en/r1/tutorials/eager/automatic_differentiation.ipynb"><img src="https://www.tensorflow.org/images/colab_logo_32px.png" />Run in Google Colab</a>
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    <a target="_blank" href="https://github.com/tensorflow/docs/blob/master/site/en/r1/tutorials/eager/automatic_differentiation.ipynb"><img src="https://www.tensorflow.org/images/GitHub-Mark-32px.png" />View source on GitHub</a>
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Note: This is an archived TF1 notebook. These are configured
to run in TF2's 
compatibility mode
but will run in TF1 as well. To use TF1 in Colab, use the
%tensorflow_version 1.x
magic.
In the previous tutorial we introduced Tensors and operations on them. In this tutorial we will cover automatic differentiation, a key technique for optimizing machine learning models.
Setup
End of explanation
x = tf.ones((2, 2))
with tf.GradientTape() as t:
  t.watch(x)
  y = tf.reduce_sum(x)
  z = tf.multiply(y, y)
# Derivative of z with respect to the original input tensor x
dz_dx = t.gradient(z, x)
for i in [0, 1]:
  for j in [0, 1]:
    assert dz_dx[i][j].numpy() == 8.0
Explanation: Gradient tapes
TensorFlow provides the tf.GradientTape API for automatic differentiation - computing the gradient of a computation with respect to its input variables. Tensorflow "records" all operations executed inside the context of a tf.GradientTape onto a "tape". Tensorflow then uses that tape and the gradients associated with each recorded operation to compute the gradients of a "recorded" computation using reverse mode differentiation.
For example:
End of explanation
x = tf.ones((2, 2))
with tf.GradientTape() as t:
  t.watch(x)
  y = tf.reduce_sum(x)
  z = tf.multiply(y, y)
# Use the tape to compute the derivative of z with respect to the
# intermediate value y.
dz_dy = t.gradient(z, y)
assert dz_dy.numpy() == 8.0
Explanation: You can also request gradients of the output with respect to intermediate values computed during a "recorded" tf.GradientTape context.
End of explanation
x = tf.constant(3.0)
with tf.GradientTape(persistent=True) as t:
  t.watch(x)
  y = x * x
  z = y * y
dz_dx = t.gradient(z, x)  # 108.0 (4*x^3 at x = 3)
dy_dx = t.gradient(y, x)  # 6.0
del t  # Drop the reference to the tape
Explanation: By default, the resources held by a GradientTape are released as soon as GradientTape.gradient() method is called. To compute multiple gradients over the same computation, create a persistent gradient tape. This allows multiple calls to the gradient() method as resources are released when the tape object is garbage collected. For example:
End of explanation
def f(x, y):
  output = 1.0
  for i in range(y):
    if i > 1 and i < 5:
      output = tf.multiply(output, x)
  return output
def grad(x, y):
  with tf.GradientTape() as t:
    t.watch(x)
    out = f(x, y)
  return t.gradient(out, x)
x = tf.convert_to_tensor(2.0)
assert grad(x, 6).numpy() == 12.0
assert grad(x, 5).numpy() == 12.0
assert grad(x, 4).numpy() == 4.0
Explanation: Recording control flow
Because tapes record operations as they are executed, Python control flow (using ifs and whiles for example) is naturally handled:
End of explanation
x = tf.Variable(1.0)  # Create a Tensorflow variable initialized to 1.0
with tf.GradientTape() as t:
  with tf.GradientTape() as t2:
    y = x * x * x
  # Compute the gradient inside the 't' context manager
  # which means the gradient computation is differentiable as well.
  dy_dx = t2.gradient(y, x)
d2y_dx2 = t.gradient(dy_dx, x)
assert dy_dx.numpy() == 3.0
assert d2y_dx2.numpy() == 6.0
Explanation: Higher-order gradients
Operations inside of the GradientTape context manager are recorded for automatic differentiation. If gradients are computed in that context, then the gradient computation is recorded as well. As a result, the exact same API works for higher-order gradients as well. For example:
End of explanation
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