Aprenda Implementing Partial Derivatives in Python

In this video, we'll learn how to compute partial derivatives of functions with multiple variables using Python. Partial derivatives are essential in optimization problems, machine learning, and data science to analyze how a function changes with respect to one variable while keeping the others constant.

1. Defining a Multivariable Function

x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2

Here, we define $x$ and $y$ as symbolic variables;
We then define the function $f(x, y) = 4x³y + 5y²$ .

2. Computing Partial Derivatives

df_dx = sp.diff(f, x)  
df_dy = sp.diff(f, y)

sp.diff(f, x) computes $\frac{∂f}{∂x}$ while treating $y$ as a constant;
sp.diff(f, y) computes $\frac{∂f}{∂y}$ while treating $x$ as a constant.

3. Evaluating Partial Derivatives at (x=1, y=2)

df_dx_val = df_dx.subs({x: 1, y: 2})  
df_dy_val = df_dy.subs({x: 1, y: 2})

The .subs({x: 1, y: 2}) function substitutes $x=1$ and $$y=2$4 into the computed derivatives;
This allows us to numerically evaluate the derivatives at a specific point.

4. Printing the Results


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import sympy

x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2

df_dx = sp.diff(f, x)  
df_dy = sp.diff(f, y)

df_dx_val = df_dx.subs({x: 1, y: 2})  
df_dy_val = df_dy.subs({x: 1, y: 2})

print("Function: f(x, y) =", f)
print("∂f/∂x =", df_dx)
print("∂f/∂y =", df_dy)
print("∂f/∂x at (1,2) =", df_dx_val)
print("∂f/∂y at (1,2) =", df_dy_val)

We print the original function, its partial derivativesw, and their evaluations at $(1,2)$ .

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Seção 3. Capítulo 8

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Can you explain what partial derivatives are in simple terms?

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1. Defining a Multivariable Function

x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2

Here, we define $x$ and $y$ as symbolic variables;
We then define the function $f(x, y) = 4x³y + 5y²$ .

2. Computing Partial Derivatives

df_dx = sp.diff(f, x)  
df_dy = sp.diff(f, y)

sp.diff(f, x) computes $\frac{∂f}{∂x}$ while treating $y$ as a constant;
sp.diff(f, y) computes $\frac{∂f}{∂y}$ while treating $x$ as a constant.

3. Evaluating Partial Derivatives at (x=1, y=2)

df_dx_val = df_dx.subs({x: 1, y: 2})  
df_dy_val = df_dy.subs({x: 1, y: 2})

The .subs({x: 1, y: 2}) function substitutes $x=1$ and $$y=2$4 into the computed derivatives;
This allows us to numerically evaluate the derivatives at a specific point.

4. Printing the Results


              12345678910111213141516
            
import sympy

x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2

df_dx = sp.diff(f, x)  
df_dy = sp.diff(f, y)

df_dx_val = df_dx.subs({x: 1, y: 2})  
df_dy_val = df_dy.subs({x: 1, y: 2})

print("Function: f(x, y) =", f)
print("∂f/∂x =", df_dx)
print("∂f/∂y =", df_dy)
print("∂f/∂x at (1,2) =", df_dx_val)
print("∂f/∂y at (1,2) =", df_dy_val)

We print the original function, its partial derivativesw, and their evaluations at $(1,2)$ .

Tudo estava claro?

Obrigado pelo seu feedback!

Seção 3. Capítulo 8

Implementing Partial Derivatives in Python

1. Defining a Multivariable Function

2. Computing Partial Derivatives

3. Evaluating Partial Derivatives at (x=1, y=2)

4. Printing the Results

1. What does the partial derivative of $f(x, y)$ with respect to $x$ represent?

2. What will `sp.diff(f, y)` return for given function?

3. If we evaluate $\frac{∂f}{∂x}$ at $(2,3)$ and the result is 24, what does this mean?

Awesome!

Implementing Partial Derivatives in Python

1. Defining a Multivariable Function

2. Computing Partial Derivatives

3. Evaluating Partial Derivatives at (x=1, y=2)

4. Printing the Results

1. What does the partial derivative of $f(x, y)$ with respect to $x$ represent?

2. What will `sp.diff(f, y)` return for given function?

3. If we evaluate $\frac{∂f}{∂x}$ at $(2,3)$ and the result is 24, what does this mean?

Implementing Partial Derivatives in Python

1. Defining a Multivariable Function

2. Computing Partial Derivatives

3. Evaluating Partial Derivatives at (x=1, y=2)

4. Printing the Results

1. What does the partial derivative of f(x,y)f(x, y)f(x,y) with respect to xxx represent?

2. What will sp.diff(f, y) return for given function?

3. If we evaluate ∂f∂x\frac{∂f}{∂x}∂x∂f​ at (2,3)(2,3)(2,3) and the result is 24, what does this mean?

Awesome!

Implementing Partial Derivatives in Python

1. Defining a Multivariable Function

2. Computing Partial Derivatives

3. Evaluating Partial Derivatives at (x=1, y=2)

4. Printing the Results

1. What does the partial derivative of f(x,y)f(x, y)f(x,y) with respect to xxx represent?

2. What will sp.diff(f, y) return for given function?

3. If we evaluate ∂f∂x\frac{∂f}{∂x}∂x∂f​ at (2,3)(2,3)(2,3) and the result is 24, what does this mean?

1. What does the partial derivative of $f(x, y)$ with respect to $x$ represent?

2. What will `sp.diff(f, y)` return for given function?

3. If we evaluate $\frac{∂f}{∂x}$ at $(2,3)$ and the result is 24, what does this mean?

1. What does the partial derivative of $f(x, y)$ with respect to $x$ represent?

2. What will `sp.diff(f, y)` return for given function?

3. If we evaluate $\frac{∂f}{∂x}$ at $(2,3)$ and the result is 24, what does this mean?