101 LearnersLast updated on July 19th, 2025
Derivative of L2 Norm
We use the derivative of the L2 norm, a powerful tool to understand how the norm changes in response to slight changes in the vector being considered. Derivatives are essential in various real-life applications, such as optimization problems and machine learning. We will now discuss the derivative of the L2 norm in detail.
What is the Derivative of L2 Norm?
We now explore the derivative of the L2 norm, which is commonly represented as ||x||₂, where x is a vector. The derivative of the L2 norm is relevant in optimization, especially in gradient descent algorithms. The L2 norm of a vector x is defined as: ||x||₂ = sqrt(x₁² + x₂² + ... + xₙ²) The derivative of the L2 norm with respect to the vector x is given by: d/dx (||x||₂) = x / ||x||₂
Derivative of L2 Norm Formula
The derivative of the L2 norm with respect to a vector x can be denoted as: d/dx (||x||₂) = x / ||x||₂ This formula applies to all vectors x ≠ 0, ensuring the denominator is not zero.
Proofs of the Derivative of L2 Norm
We can derive the derivative of the L2 norm using the chain rule and properties of the square root function. To demonstrate, we consider: ||x||₂ = sqrt(x₁² + x₂² + ... + xₙ²) By applying the chain rule, we have: d/dx (||x||₂) = d/dx (sqrt(x·x)) = (1/2) * (x·x)^(-1/2) * d/dx (x·x) Using the product rule, d/dx (x·x) = 2x: d/dx (||x||₂) = (1/2) * (x·x)^(-1/2) * 2x = x / sqrt(x·x) = x / ||x||₂ Thus, the derivative of the L2 norm is x / ||x||₂.
Higher-Order Derivatives of L2 Norm
When a function is differentiated multiple times, the results are known as higher-order derivatives. For the L2 norm, the first derivative gives us the direction of the steepest ascent or descent. Calculating higher-order derivatives of the L2 norm can be complex because further derivatives involve more intricate manipulation of vector calculus and analysis. For the first derivative of the L2 norm, we have: f′(x) = x / ||x||₂ Higher-order derivatives would require progressively more complex differentiation, often beyond basic calculus.
Special Cases:
When the vector x is the zero vector, the derivative is undefined because the L2 norm is zero, leading to division by zero.
Common Mistakes and How to Avoid Them in Derivatives of L2 Norm
Students frequently make errors when differentiating the L2 norm, often due to misunderstanding the vector calculus involved. Here are a few common mistakes and how to solve them:
Mistake 1
Not Recognizing the Vector Nature
Students may forget that the derivative of the L2 norm is a vector, not a scalar. It's important to remember that the derivative points in the direction of the vector x normalized by its norm.
Mistake 2
Forgetting the Denominator in the Formula
Sometimes, students forget to include the denominator ||x||₂, which is crucial for the correct derivative. Ensure to divide by the norm to avoid this mistake.
Mistake 3
Misapplying the Chain Rule
While differentiating, students might misapply the chain rule, leading to incorrect results. Remember to correctly apply the chain rule to the square root function.
Mistake 4
Ignoring Undefined Points
The derivative of the L2 norm is undefined at the zero vector. It's important to consider this when working with the function.
Mistake 5
Overlooking Higher-Order Derivatives
Students often assume higher-order derivatives are straightforward, but they require careful application of vector calculus and analysis techniques.
Examples Using the Derivative of L2 Norm
Problem 1
Calculate the derivative of ||x||₂ when x = (3, 4).
For x = (3, 4), the L2 norm is ||x||₂ = sqrt(3² + 4²) = 5. The derivative is given by: d/dx (||x||₂) = x / ||x||₂ = (3, 4) / 5 = (3/5, 4/5) Thus, the derivative is (3/5, 4/5).
Explanation
We calculate the L2 norm of the vector (3, 4) and then use the derivative formula to find the normalized direction of the vector.
Problem 2
A drone's position is given by the vector p(t) = (t, t²). Find the derivative of its speed, given by the L2 norm of its velocity, at t = 1.
The velocity vector is v(t) = (1, 2t). At t = 1, v(1) = (1, 2). The speed is ||v(t)||₂ = sqrt(1² + (2t)²) = sqrt(1 + 4t²). The derivative of speed with respect to t is: d/dt (||v(t)||₂) = (1 + 4t²)^(-1/2) * (8t) = 8t / sqrt(1 + 4t²) At t = 1, this becomes: 8(1) / sqrt(1 + 4(1)²) = 8 / sqrt(5) Thus, the derivative of the speed at t = 1 is 8 / sqrt(5).
Explanation
We differentiate the speed of the drone, given by the L2 norm of its velocity, using the chain rule to find how the speed changes at t = 1.
Problem 3
Derive the second derivative of the L2 norm of a vector x = (x₁, x₂).
First, find the first derivative: d/dx (||x||₂) = x / ||x||₂ To find the second derivative, differentiate the first derivative with respect to x again, which involves more complex vector calculus and considerations of directional derivatives and the Hessian matrix.
Explanation
Higher-order derivatives of the L2 norm are complex and involve advanced vector calculus techniques; they are not typically straightforward.
Problem 4
Prove: d/dx (||x||₂²) = 2x.
Let y = ||x||₂² = x·x. Differentiating, we use: d/dx (x·x) = 2x Thus, d/dx (||x||₂²) = 2x, proving the statement.
Explanation
By recognizing that ||x||₂² is the dot product of x with itself, we can use basic derivative rules of dot products to arrive at the result.
Problem 5
Solve: d/dx (||x + a||₂) where a is a constant vector.
The derivative is given by: d/dx (||x + a||₂) = (x + a) / ||x + a||₂ This formula uses the same logic as the derivative of the L2 norm for a vector.
Explanation
We treat x + a as a new vector and apply the standard derivative formula for the L2 norm, adjusting for the constant vector a.
FAQs on the Derivative of L2 Norm
1.Find the derivative of the L2 norm of a vector.
2.Can the derivative of the L2 norm be used in real-life applications?
3.Is it possible to take the derivative of the L2 norm at the zero vector?
4.What rule is used to differentiate ||x||₂²?
5.Are the derivatives of ||x||₂ and ||x||₁ the same?
Important Glossaries for the Derivative of L2 Norm
Derivative: A measure of how a function changes as its input changes, showing the rate of change or slope. L2 Norm: A measure of the magnitude of a vector, calculated as the square root of the sum of the squares of its components. Chain Rule: A fundamental rule in calculus used to differentiate the composition of functions. Vector Calculus: A branch of mathematics concerned with differentiation and integration of vector fields. Subgradient: A generalization of the derivative for functions that may not be differentiable everywhere, commonly used in optimization.
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Jaskaran Singh Saluja
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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