91Ó°ÊÓ

The power rule is a key concept in differentiation that simplifies finding the derivative of functions that are polynomials or can be expressed as power functions.
The rule states that if you have a function in the form of \( f(x) = x^n \), the derivative \( f^{\text{'}(x)} \) is given by:
\[ f^{\text{'}(x)} = n x^{n-1} \].
This means you bring the exponent \( n \) down as a coefficient and then reduce the exponent by one.
Example: For \( f(x) = x^{3} \), using the power rule, the derivative is \( f^{\text{'}(x)} = 3x^{2} \).
Applying this to \( f(x) = x^{1/2} \), we get: \( f^{\text{'}(x)} = \frac{1}{2} x^{-1/2} \).

Simplifying the derivative is an important step that makes the expression easier to understand and use.
When simplifying, the goal is to rewrite the derivative in a cleaner form. Often, this involves dealing with negative exponents or converting back to radical form.
For the function \( f^{'}(x) = \frac{1}{2} x^{-1/2} \), we have a negative exponent \( -1/2 \).
This can be expressed in terms of a positive exponent by rewriting it as a reciprocal: \( x^{-1/2} = \frac{1}{x^{1/2}} \), which further simplifies to \( \frac{1}{\sqrt{x}} \)
Thus, \( f^{'}(x) = \frac{1}{2} \frac{1}{\sqrt{x}} = \frac{1}{2\sqrt{x}} \).
This form is typically preferred as it avoids negative exponents and is more intuitive, especially when dealing with roots.

Understanding and applying exponent rules are crucial for manipulating and simplifying expressions in calculus.
Here are some fundamental exponent rules:

• \( x^a x^b = x^{a+b} \)
• \( \left( x^a \right)^b = x^{ab} \)
• \( \frac{x^a}{x^b} = x^{a-b} \)
• \( x^{-a} = \frac{1}{x^a} \)

In our example, we had \( x^{-1/2} \), which we transformed using the exponent rule for negative exponents: \( x^{-a} = \frac{1}{x^a} \).
This rule helps rewrite \( x^{-1/2} \) as \( \frac{1}{x^{1/2}} \), simplifying the derivative to \( \frac{1}{2\sqrt{x}} \).
Familiarity with these rules simplifies the process of differentiation and helps understand the underlying structure of expressions.