91Ó°ÊÓ

The power rule is a fundamental technique in calculus for differentiating functions of the form \(x^n\), where \(n\) is a constant exponent. It states: To differentiate a function \(x^n\), you multiply by the exponent \(n\) and then decrease the exponent by 1, leading to \(\frac{d}{dx}(x^n) = nx^{n-1}\). This rule simplifies the process of differentiation, making it a quick and powerful tool for calculus.

For example, if we take our function \(x^{n}\) with \(n = -\frac{4}{3}\), the power rule application gives us \(-\frac{4}{3}x^{-\frac{7}{3}}\). This step involves a systematic application of the power rule, showing its efficacy in handling non-integer and negative exponents as well as positive ones.

Exponent rules are crucial for rewriting expressions into forms that are easier to differentiate. In the original exercise, the function is represented as a fraction with a root, \(\frac{1}{\sqrt[3]{x^4}}\). This can be transformed using exponent rules:

• Roots can be expressed as fractional exponents, where \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\).
• Reciprocal rules state \(\frac{1}{x^n} = x^{-n}\).

Applying these rules, we rewrite \(\frac{1}{\sqrt[3]{x^4}}\) as \(x^{-\frac{4}{3}}\). This transformation is crucial for using the power rule effectively as it allows us to apply differentiation techniques directly.

Simplifying derivatives is the final step after they have been calculated. Once we find the derivative using the power rule, it often results in an expression that can be simplified further. In our case, the derivative \(-\frac{4}{3}x^{-\frac{7}{3}}\) needs to be expressed in a more understandable form:

• The negative exponent signifies a reciprocal, so \(x^{-\frac{7}{3}}\) becomes \(\frac{1}{x^{\frac{7}{3}}}\).
• This can be rewritten to include roots, using \(x^{\frac{7}{3}} = \sqrt[3]{x^7}\).

Thus, the simplified form of the derivative can be expressed as \(\frac{-4}{3\sqrt[3]{x^7}}\). This process not only makes the expression cleaner but also facilitates better understanding of the derivative’s behavior.