derivative of 1/square root x

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16-10-2024

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Demystifying the Derivative of 1/√x: A Step-by-Step Guide

Understanding derivatives is crucial in calculus, as they represent the instantaneous rate of change of a function. One common function encountered in calculus is 1/√x, also known as x^(-1/2). Let's dive into how to find its derivative.

Understanding the Basics

Before we tackle the derivative, let's refresh our understanding of some key concepts:

• Derivative: The derivative of a function measures how its output changes with respect to its input. For example, the derivative of the function f(x) = x² tells us how the value of x² changes as x changes.
• Power Rule: The power rule states that the derivative of x^n is n*x^(n-1). This rule is fundamental for finding derivatives of many functions.

Finding the Derivative of 1/√x

1. Rewrite the Function: First, we need to express 1/√x in a way that's easier to differentiate. Remember that √x is the same as x^(1/2). Therefore, 1/√x can be rewritten as x^(-1/2).

2. Apply the Power Rule: Now, we can apply the power rule to find the derivative:

   • d/dx [x^(-1/2)] = (-1/2) * x^(-1/2 - 1)
   • d/dx [x^(-1/2)] = (-1/2) * x^(-3/2)

3. Simplify: The final step is to simplify the expression:

   • d/dx [x^(-1/2)] = -1 / (2 * x^(3/2))
   • d/dx [x^(-1/2)] = -1 / (2√x³)

Practical Application: Finding the Tangent Line

The derivative of a function allows us to find the slope of the tangent line at any point on its graph. Let's say we want to find the slope of the tangent line to the curve y = 1/√x at the point x = 4.

1. Calculate the Derivative: We already know that the derivative of 1/√x is -1 / (2√x³).

2. Evaluate at x = 4: Substitute x = 4 into the derivative:

   • -1 / (2√(4)³) = -1 / (2√64) = -1/16.

Therefore, the slope of the tangent line to the curve y = 1/√x at the point x = 4 is -1/16.

Further Insights

• Real-World Applications: Derivatives have numerous real-world applications, including optimizing production processes, predicting stock prices, and understanding how physical quantities like velocity and acceleration change over time.
• More Complex Functions: The derivative of 1/√x is a relatively simple example. For more complex functions, other differentiation techniques like the chain rule, product rule, and quotient rule are used.

References:

• Power rule explanation
• Derivative calculator

Conclusion:

Finding the derivative of 1/√x is a fundamental exercise in calculus that highlights the power rule and its application in understanding the rate of change of functions. This example provides a solid foundation for exploring more complex derivative problems and applying them to real-world scenarios.