#Exercices

#Exercise 1: Gradient of a Simple Function

#Task

Define the function

f (x) = 3 x^{3} - 5 x^{2} + 2 x

Compute the gradient of $f (x)$ at ( x = 4 ).

Don't look at the solution before you have attempted to solve the exercise on your own.

Solution

The gradient (derivative) of $f (x)$ is given by:

\frac{d}{d x} (3 x^{3} - 5 x^{2} + 2 x) = 9 x^{2} - 10 x + 2

For ( x = 4 ):

9 \cdot 4^{2} - 10 \cdot 4 + 2 = 576 - 40 + 2 = 538

#Code Template

import torch

# Define the function f(x) = 3x^3 - 5x^2 + 2x
def function(x):
    return 3 * x**3 - 5 * x**2 + 2 * x

# Initialize the variable
x = torch.tensor(4.0, requires_grad=True)

# Compute the function value
y = function(x)

# Compute the gradient
y.backward()

# Print the gradient
print(f'The gradient of the function at x = 4 is {x.grad.item()}')

#Exercise 2: Partial Derivatives of a Simple Multivariable Function

#Task

Define the function : $f (x, y) = x^{2} + 3 x y + y^{2}$
Compute the partial derivatives of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at the point $(x, y) = (2, 3)$ .

Don't look at the solution before you have attempted to solve the exercise on your own.

Solution

1. Compute

\frac{\partial f}{\partial x}

The partial derivative of ( f ) with respect to ( x ) is:

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^{2} + 3 x y + y^{2})

Since

\frac{\partial}{\partial x} (x^{2}) = 2 x

\frac{\partial}{\partial x} (3 x y) = 3 y

and

\frac{\partial}{\partial x} (y^{2}) = 0

we get:

\frac{\partial f}{\partial x} = 2 x + 3 y

2. Compute

\frac{\partial f}{\partial y}

The partial derivative of ( f ) with respect to ( y ) is:

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^{2} + 3 x y + y^{2})

Since

\frac{\partial}{\partial y} (x^{2}) = 0

\frac{\partial}{\partial y} (3 x y) = 3 x

and

\frac{\partial}{\partial y} (y^{2}) = 2 y

we get:

\frac{\partial f}{\partial y} = 3 x + 2 y

At the point

(x, y) = (2, 3)

we have:

\frac{\partial f}{\partial x}_{(2, 3)} = 2 \cdot 2 + 3 \cdot 3 = 4 + 9 = 13

\frac{\partial f}{\partial y}_{(2, 3)} = 3 \cdot 2 + 2 \cdot 3 = 6 + 6 = 12

#Code Template

import torch

# Define the function f(x, y) = x^2 + 3xy + y^2
def function(x, y):
    return x**2 + 3 * x * y + y**2

# Initialize the variables
x = torch.tensor(2.0, requires_grad=True)
y = torch.tensor(3.0, requires_grad=True)

# Compute the function value
z = function(x, y)

# Compute the partial derivatives
z.backward()

# Print the partial derivatives
print(f'The partial derivative with respect to x at (2, 3) is {x.grad.item()}')
print(f'The partial derivative with respect to y at (2, 3) is {y.grad.item()}')