Understanding the Differentiation of x/2

In calculus, the derivative of a function provides a way to measure the sensitivity of the function to changes in its input. Understanding how to differentiate common expressions is a fundamental skill in calculus. This article focuses on the differentiation of the function x/2. We will explore the process using basic differentiation rules and the quotient rule, providing a comprehensive guide to solving this problem.

Basic Differentiation of x/2

Let's start by differentiating the function fx x/2 using the basic principles of differentiation. The derivative of x with respect to x is 1, and since 1/2 is a constant, we can factor it out.

The mathematical steps are as follows:

[f(x)  frac{d}{dx} left( frac{x}{2} right)  frac{1}{2} cdot frac{d}{dx} x  frac{1}{2} cdot 1  frac{1}{2}]

Thus, the derivative of x/2 with respect to x is 1/2.

Verifying with Differentiation Rules

We can also verify this result using the quotient rule, which is a standard method for differentiating functions of the form u/v. The quotient rule states:

Numerator'/Denominator - Denominator'Numerator / denominator^2

In our case, u x and v 2. Therefore:

[ frac{d}{dx} left( frac{x}{2} right)  frac{(1)(2) - (x)(0)}{2^2}  frac{2}{4}  frac{1}{2} ]

The derivative is again 1/2, confirming our initial result.

Alternative Method

Another straightforward method involves considering the function as a product of a constant and a variable. Let's represent it as:

This can be rewritten as:

[ y  frac{1}{2} times frac{d}{dx} x ]

Solving this step-by-step:

[ y  frac{1}{2} times 1  frac{1}{2} ]

This again gives us the derivative as 1/2.

Conclusion

Whether we use the basic differentiation rule, the quotient rule, or consider the function as a product of constants and variables, the result remains the same. The derivative of x/2 with respect to x is indeed 1/2.

This article provides a thorough explanation of the differentiation of x/2 using different methods. It is crucial for anyone studying calculus to understand these basic principles and how to apply them to various functions. If you have any questions or need further practice, feel free to comment or reach out for additional help.

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