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 ∂x∂f while treating y as a constant;sp.diff(f, y)computes ∂y∂f 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 $$y=2$4 into the computed derivatives; - This allows us to numerically evaluate the derivatives at a specific point.
4. Printing the Results
12345678910111213141516import sympy x, y = sp.symbols('x y') f = 4*x**3*y + 5*y**2 df_dx = sp.diff(f, x) df_dy = sp.diff(f, y) df_dx_val = df_dx.subs({x: 1, y: 2}) df_dy_val = df_dy.subs({x: 1, y: 2}) print("Function: f(x, y) =", f) print("∂f/∂x =", df_dx) print("∂f/∂y =", df_dy) print("∂f/∂x at (1,2) =", df_dx_val) print("∂f/∂y at (1,2) =", df_dy_val)
- We print the original function, its partial derivativesw, and their evaluations at (1,2).
1. What does the partial derivative of f(x,y) with respect to x represent?
2. What will sp.diff(f, y) return for given function?
f(x,y)=x2y+3y23. If we evaluate ∂x∂f at (2,3) and the result is 24, what does this mean?
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In this video, we'll learn how to compute partial derivatives of functions with multiple variables using Python. Partial derivatives are essential in optimization problems, machine learning, and data science to analyze how a function changes with respect to one variable while keeping the others constant.
1. Defining a Multivariable Function
x, y = sp.symbols('x y')
f = 4*x**3*y + 5*y**2
- Here, we define x and y as symbolic variables;
- We then define the function f(x,y)=4x3y+5y2.
2. Computing Partial Derivatives
df_dx = sp.diff(f, x)
df_dy = sp.diff(f, y)
sp.diff(f, x)computes ∂x∂f while treating y as a constant;sp.diff(f, y)computes ∂y∂f while treating x as a constant.
3. Evaluating Partial Derivatives at (x=1, y=2)
df_dx_val = df_dx.subs({x: 1, y: 2})
df_dy_val = df_dy.subs({x: 1, y: 2})
- The
.subs({x: 1, y: 2})function substitutes x=1 and $$y=2$4 into the computed derivatives; - This allows us to numerically evaluate the derivatives at a specific point.
4. Printing the Results
12345678910111213141516import 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).
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?
f(x,y)=x2y+3y23. If we evaluate ∂x∂f at (2,3) and the result is 24, what does this mean?
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