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The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. These are functions that are formed by applying one function to the result of another function. In simpler terms, the chain rule helps you differentiate functions that are "nested," or within other functions.

For example, if you have a function like \[ f(x) = (x^5 - x + 1)^{-9} \], you're dealing with a composite because one function, say \( u(x) = x^5 - x + 1 \), is inside another function, \(u^n = u^{-9} \). The chain rule formula is:

• \( y' = u' \, n \, u^{n-1} \)

Here, you first take the derivative of the outer function \(u^n\), assuming \(u\) is a constant, and then multiply it by the derivative of \(u\). This combination allows you to handle complex derivatives step by step.

The power rule is one of the simplest and most frequently used rules in differentiation. It applies to functions of the form \( x^n \), where \( n \) is any real number. According to the power rule, the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

In the problem at hand, where the expression \((x^5 - x + 1)^{-9}\) is treated as the function \(u(x)^{-9}\), it simplifies the differentiation process.

• The formula \( n\cdot u^{n-1} \) means you multiply \( u \) raised to the new power \( n-1 \) by the original power \( n \).

When dealing with functions like \(x^5 - x + 1\), apply the power rule by differentiating each term separately first.

Derivative calculations involve identifying the appropriate rule or combination of rules to apply, simplifying terms, and computing the necessary derivatives. In this example, the derivative of \( f(x) = \frac{1}{(x^5 - x + 1)^9} \) was approached by first rewriting it in a form suitable for the chain and power rules.

Steps to calculating the derivative include:

• Identifying composite functions to use the chain rule.
• Applying the power rule to simplify derivatives of terms like \( x^n \).
• Finding the derivative of the inner function, \( x^5 - x + 1 \), by treating each term separately, yielding \( 5x^4 - 1 \).
• Combining these results using multiplication to complete the chain rule calculation.

The final result is a product of all these derivatives, simplified to provide a clean answer, \(-9(5x^4 - 1)(x^5 - x + 1)^{-10}\). Practice these steps regularly to become proficient in derivative calculations.