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(x^2+y^2)^1/2 Derivative

4 min read Jun 17, 2024
(x^2+y^2)^1/2 Derivative

Understanding the Derivative of (x^2 + y^2)^1/2

The expression (x² + y²)¹/² represents the distance formula in two dimensions. It calculates the distance between the origin (0, 0) and a point (x, y) in the Cartesian plane.

To find the derivative of this expression, we need to consider whether x and y are independent variables or dependent variables. Let's explore both cases:

Case 1: x and y are independent variables

If x and y are independent variables, we can treat them separately when differentiating. The derivative is then:

∂/∂x [(x² + y²)¹/²] = x / (x² + y²)¹/²

∂/∂y [(x² + y²)¹/²] = y / (x² + y²)¹/²

These derivatives represent the rate of change of the distance with respect to x and y, respectively.

Case 2: y is a function of x (y = f(x))

When y is a function of x, we need to use the chain rule for differentiation. Here's how it works:

  1. Rewrite the expression: (x² + [f(x)]²)¹/²
  2. Apply the chain rule:
    • The derivative of the outer function (x² + y²)¹/² is (1/2)(x² + y²)⁻¹/².
    • The derivative of the inner function (x² + [f(x)]²) is 2x + 2f(x)f'(x).
  3. Combine the derivatives: d/dx [(x² + y²)¹/²] = (1/2)(x² + y²)⁻¹/² * (2x + 2f(x)f'(x)) d/dx [(x² + y²)¹/²] = (x + f(x)f'(x)) / (x² + y²)¹/²

This derivative represents the rate of change of the distance with respect to x when y is a function of x.

Applications of the Derivative

The derivative of (x² + y²)¹/² has applications in various fields, including:

  • Geometry: Calculating the length of a curve.
  • Physics: Determining the velocity of a moving object.
  • Calculus: Finding the tangent line to a curve.

Important Notes:

  • The derivative of (x² + y²)¹/² is a vector that points in the direction of the gradient of the function.
  • The magnitude of the derivative represents the rate of change of the distance.
  • The derivative is zero when the distance is at a maximum or minimum.

By understanding the derivative of (x² + y²)¹/², we can gain insights into the properties of this fundamental mathematical function and its applications in different areas of study.

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