91Ó°ÊÓ

The product rule is a fundamental tool in calculus used for finding the derivative of a product of two functions. If we have two functions of x, let's call them \( u(x) \) and \( v(x) \), their product is \( u(x)v(x) \), and according to the product rule, the derivative of this product is:

• \( (uv)' = u'v + uv' \)

This rule is essential because derivatives of products are not simply the products of derivatives.
For example, in the given problem, the function is rewritten as a product: \( f(x) = (x^2 + 1)^{1/2} \cdot (x^{1/2} + 1)^3 \).
Here, we differentiate \( (x^2 + 1)^{1/2} \) and \( (x^{1/2} + 1)^3 \) separately and then apply the product rule formula.
By doing so, we separate the process into manageable parts, allowing us to accurately find the derivative of the entire function.

Simplifying functions is often a necessary step in solving complex calculus problems. It involves rewriting a function in a form that is easier to handle mathematically.
In the original exercise, the function \( f(x) \) was simplified from a more complex radical expression to a product of powers: \( f(x) = (x^2 + 1)^{1/2} \cdot (x^{1/2} + 1)^3 \).
This transformation made applying the product rule much more straightforward.

• Simplifying functions can include factoring expressions to reveal useful product forms.
• Rewriting radicals as exponents makes them easier to differentiate.
• It helps in understanding the structure of a function, allowing for more effective manipulation.

Understanding how to simplify functions will aid in both finding and simplifying derivatives.

Creating common denominators is a technique used in algebra to combine fractions, and it becomes particularly useful during the simplification of derivatives.
When working with derivatives, especially those involving fractions, aligning terms under a common denominator helps in combining them neatly into a single expression.
In our problem, common denominators were used to tidy the expression of \( f'(x) \) after applying the product rule:

• The terms \( \frac{x(x^{1/2} + 1)^3}{\sqrt{x^2 + 1}} \) and \( \frac{(3/2)(x^{1/2} + 1)^2}{\sqrt{x} \sqrt{x^2 + 1}} \) share common elements, \( \sqrt{x^2 + 1} \).
• By rewriting fractions to share the same denominator, we simplify the expression to one more easily manipulated or interpreted later on.

Creating common denominators is not just helpful for simplifying, but it also clarifies the results, providing a cleaner and more understandable final derivative.