Derivative of yx√x from First Principles: A Comprehensive Guide

Understanding how to find derivatives from first principles is a fundamental skill in calculus. In this guide, we will explore the derivative of yx√x using the first principles approach. This method involves a step-by-step process that helps in comprehending the underlying mathematical concepts and techniques.

Step-by-Step Derivation Using First Principles

To find the derivative of the function yx√x from first principles, we start with the definition of the derivative. The derivative of a function f(x) is given by:

1. Define the Function

Let f(x) x√x.

2. Calculate f(x h)

We need to find f(x h) which is:

f(x h) (x h)√(x h)

3. Apply the Derivative Formula

Now we substitute f(x h) and f(x) into the derivative formula:

f'(x) lim_{h to 0} (f(x h) - f(x))/h

Substituting the expressions, we get:

f'(x) lim_{h to 0} ((x h)√(x h) - x√x)/h

4. Simplify the Expression

This can be simplified further as:

f'(x) lim_{h to 0} (x√(x h) - x√x h√(x h))/h

Splitting the expression:

f'(x) lim_{h to 0} (x(√(x h) - √x) h√(x h))/h

This simplifies to:

f'(x) lim_{h to 0} (x*(√(x h) - √x)/h h*√(x h)/h)

Which further simplifies to:

f'(x) lim_{h to 0} (x*(√(x h) - √x)/h √(x h))

5. Simplify the Fraction

To simplify the fraction, we multiply the numerator and the denominator by the conjugate of the numerator:

(√(x h) - √x) / (h) * ( (√(x h) √x) / ( (√(x h) √x) )

This gives us:

((x h) - x) / (h(√(x h) √x)) h / (h(√(x h) √x)) 1 / (√(x h) √x)

Note that as h approaches 0, √(x h) approaches √x, so we can simplify the expression further:

f'(x) x * (1 / (2√x)) 0 x / (2√x) 1 / (2√x)

Conclusion

The derivative of yx√x is:

dy/dx 1 / (2√x)

Why Learn Derivatives from First Principles?

Derivatives from first principles provide a rigorous foundation in calculus, helping students understand the underlying concepts more deeply. For many, understanding these basics is crucial for advanced calculus and related fields in mathematics, engineering, and science.

Acknowledging Common Pitfalls

It's important to note that some students might struggle with this concept due to a lack of understanding of basic rules of differentiation such as the Addition Rule and the Power Rule, or a lack of proficiency in algebra. However, mastering these fundamentals is essential for success in higher-level mathematics.

Conclusion

The derivative of yx√x from first principles is a valuable exercise for strengthening analytical skills and understanding. By following the steps outlined in this guide, you can confidently tackle similar problems and build a strong mathematical foundation.