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bookImplementing 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 xx and yy as symbolic variables;
  • We then define the function f(x,y)=4x3y+5y2f(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 ∂f∂x\frac{∂f}{∂x} while treating yy as a constant;
  • sp.diff(f, y) computes ∂f∂y\frac{∂f}{∂y} while treating xx 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=1x=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)
copy
  • We print the original function, its partial derivativesw, and their evaluations at (1,2)(1,2).

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

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

f(x,y)=x2y+3y2f(x, y) = x²y + 3y²

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

question mark

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

Select the correct answer

question mark

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

f(x,y)=x2y+3y2f(x, y) = x²y + 3y²

Select the correct answer

question mark

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

Select the correct answer

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Sektion 3. Kapitel 8

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bookImplementing Partial Derivatives in Python

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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 xx and yy as symbolic variables;
  • We then define the function f(x,y)=4x3y+5y2f(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 ∂f∂x\frac{∂f}{∂x} while treating yy as a constant;
  • sp.diff(f, y) computes ∂f∂y\frac{∂f}{∂y} while treating xx 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=1x=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)
copy
  • We print the original function, its partial derivativesw, and their evaluations at (1,2)(1,2).

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

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

f(x,y)=x2y+3y2f(x, y) = x²y + 3y²

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

question mark

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

Select the correct answer

question mark

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

f(x,y)=x2y+3y2f(x, y) = x²y + 3y²

Select the correct answer

question mark

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

Select the correct answer

Var alt klart?

Hvordan kan vi forbedre det?

Tak for dine kommentarer!

Sektion 3. Kapitel 8
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