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In the previous tutorial we introduced `Tensor`

s and operations on them. In this tutorial we will cover automatic differentiation, a key technique for optimizing machine learning models.

## Setup

```
from __future__ import absolute_import, division, print_function, unicode_literals
import tensorflow as tf
```

## 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:

```
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
```

You can also request gradients of the output with respect to intermediate values computed during a "recorded" `tf.GradientTape`

context.

```
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
```

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:

```
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
```

### Recording control flow

Because tapes record operations as they are executed, Python control flow (using `if`

s and `while`

s for example) is naturally handled:

```
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
```

### 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:

```
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
```

## Next Steps

In this tutorial we covered gradient computation in TensorFlow. With that we have enough of the primitives required to build and train neural networks.

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```