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In calculus, derivative calculations are essential as they help us understand how functions change. When we calculate a derivative, we're finding how fast a function's value changes at any given point. This can show us trends like growth or decline and is vital in many fields like physics, engineering, and economics.
To start calculating a derivative, one of the most common methods used is applying derivative rules, among which the power rule and chain rule are particularly popular.
By carefully using these rules, you can determine the slope of a tangent line to the curve of a function at any point, which is precisely what the derivative represents. This fundamental tool assists in solving complex problems involving rates of change.

The power rule is a quick and easy way to find the derivative of functions in the form of \(x^n\), where \(n\) is any real number. Here's how it works: if you have a function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
This rule is an essential building block in calculus:

• Reduces calculation time for polynomials.
• Simple to apply across various functions.
• Forms basis for more complex rules.

Using the power rule, you can quickly determine derivatives of polynomial expressions, like in our exercise.The power rule lets us find the derivative of \(x^{-2}\), resulting in \(-2x^{-3}\), and for a more complex term like \(2x^3 + x\), making its derivative \(6x^2 + 1\).
These easy calculations show the utility and efficiency of the power rule in simplifying derivative calculations.

The chain rule is crucial for finding the derivative of composite functions; that is, when one function is nested within another.
This rule is like a chain reaction, hence its name! It involves deriving the outer function and then multiplying it by the derivative of the inner function.
Essentially, it follows this mathematical format: \( f'(x) = g'(h(x)) \times h'(x) \).

• Must identify inner and outer functions.
• Apply the derivative to each individually.
• Multiply the derivatives together.

In our exercise, \(f(x) = (2x^3 + x)^{-2}\) is a composite function where \(h(x) = 2x^3 + x\) and the outer function \(g(x) = x^{-2}\). The chain rule helps us find \(f'(x)\) by multiplying the derivative of the nesting \(h(x)\) by \(-2(2x^3 + x)^{-3}\), yielding the complete derivative of the function.

After deriving a function, it’s important to simplify the obtained derivative. Simplification makes the expression concise and manageable, which is helpful for further analysis or application.
Here are some steps:

• Factor out common terms.
• Reduce complex fractions or expressions.
• Write in an easily interpretable format.

In the solution given for the exercise, we simplified by factoring out \((2x^3 + x)^{-3}\). This simplification made our derivative appear as \(-2(6x^2 + 1)(2x^3 + x)^{-3}\), much clearer and easier to use in downstream applications.
Being capable of simplifying derivatives efficiently not only makes calculations easier but also enhances your understanding of the mathematical relationships involved.