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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:
 <a href="http
Step1:  <!--Empty Space for separating topics-->
<h2 id="Derivative">Derivatives</h2>
Let us create the tensor <code>x</code> and set the parameter <code>requires_grad</code> to true because you are going to take the derivative of the tensor.
Step2:  Then let us create a tensor according to the equation $ y=x^2 $.
Step3:  Then let us take the derivative with respect x at x = 2
Step4:  The preceding lines perform the following operation
Step5:  The function is in the following form
Step6:  Double-click <b>here</b> for the solution.
<!-- 
x = torch.tensor(1.0, requires_grad=True)
y = 2 * x ** 3 + x
y.backward()
print("The derivative result
Step7:  This is equivalent to the following
Step8:  the expression is given by
Step9:  The equation is given by
Step10:  We can plot the function  and its derivative
Step11:  The orange line is the slope of the blue line at the intersection point, which is the derivative of the blue line.
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The <b>relu</b> activation function is an essential function in neural networks. We can take the derivative as follows
Step12:  <!--Empty Space for separating topics-->
<h3>Practice</h3>
Try to determine partial derivative  $u$ of the following function where $u=2$ and $v=1$
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Python Code:
# These are the libraries will be useing for this lab.
import torch 
import matplotlib.pylab as plt
import torch.functional as F
Explanation: <a href="http://cocl.us/NotebooksPython101">
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<h1>Differentiation in PyTorch</h1>
<h2>Table of Contents</h2>
<p>In this lab, you will learn the basics of differentiation.</p>
<ul>
    <li><a href="#Derivative">Derivatives</a></li>
    <li><a href="#Partial_Derivative">Partial Derivatives</a></li>
</ul>
<p>Estimated Time Needed: <strong>25 min</strong></p>
<hr>
<h2>Preparation</h2>
The following are the libraries we are going to use for this lab.
End of explanation
# Create a tensor x
x = torch.tensor(2.0, requires_grad = True)
print("The tensor x: ", x)
Explanation: <!--Empty Space for separating topics-->
<h2 id="Derivative">Derivatives</h2>
Let us create the tensor <code>x</code> and set the parameter <code>requires_grad</code> to true because you are going to take the derivative of the tensor.
End of explanation
# Create a tensor y according to y = x^2
y = x ** 2
print("The result of y = x^2: ", y)
Explanation: Then let us create a tensor according to the equation $ y=x^2 $.
End of explanation
# Take the derivative. Try to print out the derivative at the value x = 2
y.backward()
print("The dervative at x = 2: ", x.grad)
Explanation: Then let us take the derivative with respect x at x = 2
End of explanation
# Calculate the y = x^2 + 2x + 1, then find the derivative 
x = torch.tensor(2.0, requires_grad = True)
y = x ** 2 + 2 * x + 1
print("The result of y = x^2 + 2x + 1: ", y)
y.backward()
print("The dervative at x = 2: ", x.grad)
Explanation: The preceding lines perform the following operation: 
$\frac{\mathrm{dy(x)}}{\mathrm{dx}}=2x$
$\frac{\mathrm{dy(x=2)}}{\mathrm{dx}}=2(2)=4$
Let us try to calculate the derivative for a more complicated function.
End of explanation
# Practice: Calculate the derivative of y = 2x^3 + x at x = 1
# Type your code here
Explanation: The function is in the following form:
$y=x^{2}+2x+1$
The derivative is given by:
$\frac{\mathrm{dy(x)}}{\mathrm{dx}}=2x+2$
$\frac{\mathrm{dy(x=2)}}{\mathrm{dx}}=2(2)+2=6$
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<h3>Practice</h3>
Determine the derivative of $ y = 2x^3+x $ at $x=1$
End of explanation
# Calculate f(u, v) = v * u + u^2 at u = 1, v = 2
u = torch.tensor(1.0,requires_grad=True)
v = torch.tensor(2.0,requires_grad=True)
f = u * v + u ** 2
print("The result of v * u + u^2: ", f)
Explanation: Double-click <b>here</b> for the solution.
<!-- 
x = torch.tensor(1.0, requires_grad=True)
y = 2 * x ** 3 + x
y.backward()
print("The derivative result: ", x.grad)
 -->
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<h2 id="Partial_Derivative">Partial Derivatives</h2>
We can also calculate <b>Partial Derivatives</b>. Consider the function: $f(u,v)=vu+u^{2}$
Let us create <code>u</code> tensor, <code>v</code> tensor and  <code>f</code> tensor
End of explanation
# Calculate the derivative with respect to u
f.backward()
print("The partial derivative with respect to u: ", u.grad)
Explanation: This is equivalent to the following: 
$f(u=1,v=2)=(2)(1)+1^{2}=3$
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Now let us take the derivative with respect to <code>u</code>:
End of explanation
# Calculate the derivative with respect to v
print("The partial derivative with respect to u: ", v.grad)
Explanation: the expression is given by:
$\frac{\mathrm{\partial f(u,v)}}{\partial {u}}=v+2u$
$\frac{\mathrm{\partial f(u=1,v=2)}}{\partial {u}}=2+2(1)=4$
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Now, take the derivative with respect to <code>v</code>:
End of explanation
# Calculate the derivative with multiple values
x = torch.linspace(-10, 10, 10, requires_grad = True)
Y = x ** 2
y = torch.sum(x ** 2)
Explanation: The equation is given by:
$\frac{\mathrm{\partial f(u,v)}}{\partial {v}}=u$
$\frac{\mathrm{\partial f(u=1,v=2)}}{\partial {v}}=1$
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Calculate the derivative with respect to a function with multiple values as follows. You use the sum trick to produce a scalar valued function and then take the gradient:
End of explanation
# Take the derivative with respect to multiple value. Plot out the function and its derivative
y.backward()
plt.plot(x.detach().numpy(), Y.detach().numpy(), label = 'function')
plt.plot(x.detach().numpy(), x.grad.detach().numpy(), label = 'derivative')
plt.xlabel('x')
plt.legend()
plt.show()
Explanation: We can plot the function  and its derivative
End of explanation
import torch.nn.functional as F
# Take the derivative of Relu with respect to multiple value. Plot out the function and its derivative
x = torch.linspace(-3, 3, 100, requires_grad = True)
Y = F.relu(x)
y = Y.sum()
y.backward()
plt.plot(x.detach().numpy(), Y.detach().numpy(), label = 'function')
plt.plot(x.detach().numpy(), x.grad.detach().numpy(), label = 'derivative')
plt.xlabel('x')
plt.legend()
plt.show()
Explanation: The orange line is the slope of the blue line at the intersection point, which is the derivative of the blue line.
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The <b>relu</b> activation function is an essential function in neural networks. We can take the derivative as follows:
End of explanation
# Practice: Calculate the derivative of f = u * v + (u * v) ** 2 at u = 2, v = 1
# Type the code here
Explanation: <!--Empty Space for separating topics-->
<h3>Practice</h3>
Try to determine partial derivative  $u$ of the following function where $u=2$ and $v=1$: $ f=uv+(uv)^2$
End of explanation
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