91Ó°ÊÓ

The power rule is a fundamental technique in calculus for finding the derivative of power functions. A power function has the form \(x^n\), where \(n\) is any real number. The power rule simplifies the process by providing a simple formula: When differentiating \(x^n\), just multiply by the exponent \(n\) and reduce the exponent by one, giving us \(\frac{d}{dx}[x^n] = n*x^{n-1}\).

This rule is essential for tackling almost any function that involves powers, including both whole numbers and fractions as exponents. In our example, the function \(f(x) = \sqrt{x} - 6 \sqrt[3]{x}\) can be expressed in terms of exponents as \(f(x) = x^{1/2} - 6x^{1/3}\).

Using the power rule, we differentiate each term separately:

• The derivative of \(x^{1/2}\) is \(\frac{1}{2}x^{-1/2}\)
• The derivative of \(-6x^{1/3}\) is \(-6 * \frac{1}{3}x^{-2/3} = -2x^{-2/3}\)

This technique greatly reduces the complexity of finding derivatives for polynomial expressions.

Differentiation is the process of finding the derivative of a function. The derivative is a measure of how a function changes as its input changes — it's essentially the rate of change or the slope of the function at any given point.

In practical terms, to differentiate a function, you apply specific rules that correspond to the types of terms in the function. Here, we focus on power functions, which often appear in various forms throughout calculus and physics. By utilizing the power rule discussed earlier, we can find derivatives quickly and efficiently.
Differentiation is a key concept because it allows us to analyze the behavior of functions. Let's remember:

• Finding tangents of curves
• Determining increasing or decreasing trends
• Solving real-world problems involving rates of change, such as speed or growth

For \(f(x) = x^{1/2} - 6x^{1/3}\), differentiation provides insights into how this function behaves as \(x\) changes.

Simplifying algebraic expressions is crucial in making our results more comprehensible and usable. After calculating derivatives, it's common practice to present them in their simplest form to help with further analysis or solutions.

Consider the derivative we found: \(f'(x) = \frac{1}{2}x^{-1/2} - 2x^{-2/3}\). To simplify:

• The term \(\frac{1}{2}x^{-1/2}\) can be rewritten as \(\frac{1}{2\sqrt{x}}\). Remember, negative exponents indicate a reciprocal, so \(x^{-1/2}\) is \(\frac{1}{\sqrt{x}}\).
• The term \(-2x^{-2/3}\) becomes \(\frac{-2}{x^{2/3}}\), using similar logic: \(x^{-2/3}\) is \(\frac{1}{x^{2/3}}\).

Writing expressions in these simplified formats makes it easier to evaluate their behavior at specific points or integrate into broader mathematical solutions.