91Ó°ÊÓ

The Power Rule is a fundamental part of calculus and differentiation. It gives us a straightforward way to differentiate functions of the form \( x^n \) where \( n \) is any real number. Here's how it works:
\[ \frac{d}{dx} x^n = nx^{n-1} \]
This rule tells us to multiply the function by its exponent \( n \) and then subtract one from the exponent. It's very useful for quickly finding the derivative of polynomials and similar expressions. Let's see it in action with an example:
Consider the function \( y = x^{3/2} \).
Using the Power Rule, we differentiate it as:
\[ \frac{dy}{dx} = \frac{3}{2} x^{(3/2) - 1} \]
This simplifies to \[ \frac{dy}{dx} = \frac{3}{2} x^{1/2} \].
Easy, right? Once you practice with a few examples, applying this rule will become second nature.

Simplifying functions makes differentiation much easier. This involves rewriting a given function in a simpler form before applying differentiation rules.
Take the function given in the exercise:
\[ y = x \sqrt{x} \].
This can be rewritten using exponent rules:
\[ y = x \left( x^{1/2} \right) \] which simplifies to:\[ y = x^{3/2} \]
By rewriting the function this way, it becomes clear that we can use the Power Rule to differentiate it. Simplifying functions before differentiating not only makes the process more manageable but also helps avoid common mistakes.

Exponents are a way to express repeated multiplication. For instance, \( x^n \) means multiplying \( x \) by itself \( n \) times. Understanding exponents is crucial for both simplifying functions and applying differentiation rules.
Here are some key points about exponents:

• \[ x^1 = x \] – Any number raised to the power of 1 is itself.
• \[ x^0 = 1 \] – Any number raised to the power of 0 is 1.
• \[ x^a \cdot x^b = x^{a + b} \] – When multiplying like bases, add the exponents.
• \[ \left( x^a \right)^b = x^{a \cdot b} \] – When raising a power to another power, multiply the exponents.

In our exercise, knowing \( \sqrt{x} = x^{1/2} \) allowed us to rewrite and later differentiate the function. By mastering how exponents work, you can effectively handle polynomial expressions and their derivatives.