## 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:

**Rewrite the expression:**(x² + [f(x)]²)¹/²**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).

**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.