91Ó°ÊÓ

When it comes to finding the derivative of a function, the power rule is one of the simplest and most frequently used tools. The power rule states that if you have a function in the form of \(f(x) = x^n\), where \(n\) is any real number, the derivative is given by:

• \(f'(x) = n \times x^{n-1}\)

To apply it, you basically bring the exponent down as a coefficient and reduce the exponent by one. For instance, if \(f(x) = x^3\), then \(f'(x) = 3x^2\). This method makes it quick and simple to differentiate polynomial functions.
In the given problem, after simplifying the expression to \(y = x^{1/18}\), the power rule becomes applicable. By differentiating, we get:

• \(\frac{dy}{dx} = \frac{1}{18} x^{1/18 - 1}\)
• \(\frac{dy}{dx} = \frac{1}{18} x^{-17/18}\)

This makes it clear how straightforward the power rule can be when applied correctly.

Exponent properties are crucial in simplifying expressions before differentiation. These properties govern how exponents are handled in multiplication, division, and exponentiation. Here are a few key properties:

• \(a^m \times a^n = a^{m+n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\)
• \((a^m)^n = a^{m \times n}\)
• \(a^{-m} = \frac{1}{a^m}\)

In the exercise, we used these properties to simplify the expression \(y = \sqrt[3]{ \frac {x^2}{x^{3/2}} }\). By converting the radicals into fractional exponents, we combined and simplified them as follows:

• \( \frac{x^{2/3}}{x^{1/2}} = x^{2/3 - 1/2} = x^{1/6} \)

Understanding these rules makes it easier to work with complex algebraic expressions and prepares them for differentiation.

Fractional exponents, also known as rational exponents, provide a way to express roots as powers. A fractional exponent \(a^{m/n}\) is equivalent to the \(n\)-th root of \(a\) raised to the power \(m\). For example:

• \(a^{1/n} = \sqrt[n]{a}\)
• \(a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\)

These notations make it easier to manipulate expressions involving roots. For the given exercise, the initial form \(y = \sqrt[3]{ \frac {\sqrt[3]{x^2}}{\sqrt{x^3}} }\) was converted to a fractional exponent notation:

• \(y = \frac{x^{2/3}}{x^{1/2}}\)

By consolidating these, we simplified the problem down to \(y = x^{1/18}\), making the application of the power rule straightforward. This highlights the significant utility of fractional exponents in calculus.