Partial Derivative of the Function

Partial derivative

If we are talking about the functions with multiple arguments, we must consider the concept of partial derivative. A partial derivative measures how a function changes with respect to one of its variables while keeping the other variables constant. It is a concept used in multivariable calculus to analyze the rate of change of a function in multiple dimensions.
If we have a function y = f(x1, x2, ..., xn) the partial derivative with respect to argument xi is defined as follows:


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import sympy as sp

# Define the variables
x, y = sp.symbols('x y')

# Define the function
f = x**3 + 2*x*y + y**2 + x - 3*y

# Calculate the partial derivative with respect to x
df_dx = sp.diff(f, x)

# Calculate the partial derivative with respect to y
df_dy = sp.diff(f, y)

# Define the point at which to evaluate the partial derivatives
point = {x: 2, y: 3}

# Evaluate the partial derivatives at the given point
df_dx_value = df_dx.subs(point)
df_dy_value = df_dy.subs(point)

# Print the partial derivatives
print(f'Partial derivative with respect to x: {df_dx}')
print(f'Partial derivative with respect to y: {df_dy}')

# Print the partial derivatives at the given point
print(f'Partial derivative with respect to x at point {point}: {df_dx_value}')
print(f'Partial derivative with respect to y at point {point}: {df_dy_value}')

Higher-order derivatives

We can also take derivatives and partial derivatives of higher orders. For example, the second (partial) derivative is defined as the (partial) derivative of the first (partial) derivative of a function, and so on:

We can use sympy library to calculate derivative of the second order too:


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import sympy as sp

# Define the variable
x, y = sp.symbols('x y')

# Define the function
f = 2*x**3 + sp.sin(x*y)

# Calculate the second-order derivative
d2f_dx2 = sp.diff(f, x, 2)

# Print the second-order derivative
print(f'Second-order partial derivative with respect to x is: {d2f_dx2}')

We used d2f_dx2 = sp.diff(f, x, 2) to calculate the derivative of the second order of the function f with respect to argument x.

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Seção 3. Capítulo 2