derivative of square root x 2

Guri Bee

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21-10-2024

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Unveiling the Derivative of √x²: A Step-by-Step Guide

Understanding derivatives is crucial in calculus, enabling us to analyze the rate of change of functions. Today, we'll explore the derivative of the function √x² (which is simply x for positive values of x).

What is the Derivative of √x²?

The derivative of √x² is simply 1.

Why is this so?

Here's a breakdown:

1. Simplifying √x²: Since the square root and squaring operations cancel each other out for positive values of x, we can simplify √x² to x.

2. The Power Rule: The derivative of a power function x^n is nx^(n-1). In this case, our function is x^1, so its derivative is 1x^(1-1) = 1x^0 = 1.

Practical Application:

Understanding the derivative of √x² has applications in various fields:

• Physics: Finding the velocity of an object with constant acceleration.
• Engineering: Analyzing the rate of change of a variable in a system.
• Economics: Determining the marginal cost of production.

Visualizing the Derivative:

The derivative of √x² is a constant function, meaning its value is always 1. This translates to a horizontal line on the graph of the derivative.

Code Example:

Let's use Python to demonstrate the derivative:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 5, 100)
y = np.sqrt(x**2)

plt.plot(x, y, label="√x²")
plt.plot(x, np.ones(len(x)), label="Derivative")
plt.xlabel("x")
plt.ylabel("y")
plt.title("Derivative of √x²")
plt.legend()
plt.show()

This code generates a plot showing the function √x² and its derivative, which is a horizontal line at y=1.

Note: While the derivative of √x² is 1 for positive values of x, it's important to remember that the function √x² is not differentiable at x=0. This is because the function has a sharp corner at that point.

In conclusion: The derivative of √x² is a simple yet powerful concept with practical applications in various fields. By understanding the power rule and its simplification, we can effectively analyze the rate of change of this function.