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The derivative of a function represents how the function changes as its input changes. It provides the rate of change or the slope of the function at any given point. In calculus, this is a foundational concept used to understand the behavior of functions.

The derivative formula, especially for polynomial functions, uses the *power rule*. For a term in the form of \(ax^n\), the derivative is obtained by multiplying the exponent \(n\) by the coefficient \(a\) and decreasing the exponent by 1. Therefore, the derivative is expressed as \(f'(x) = anx^{n-1}\).

For example, if we have \(f(x) = 22x^{-3}\) and \(f(x) = -22x^3\), the derivatives are calculated as follows:

• For \(22x^{-3}\), derivative is \(-66x^{-4}\).
• For \(-22x^3\), derivative is \(-66x^2\).

Combining these results gives the full derivative: \(\frac{df}{dx} = -66x^{-4} - 66x^2\). This derivative tells us how \(f(x)\) changes with \(x\), capturing increases or decreases at specific values.

Integration is essentially the reverse process of differentiation. While a derivative gives us the rate of change, an integral allows us to determine the accumulated change, or area under a curve, over an interval.

The indefinite integral involves finding a function whose derivative is the integrand. We often use the *power rule* for integration, which is critical in solving polynomial integrals. For a function \(ax^n\), its indefinite integral is \(\frac{a}{n+1}x^{n+1} + C\), where \(C\) is the constant of integration.

For the derivative \(-66x^{-4} - 66x^2\), do the following integrations:

• Integrating \(-66x^{-4}\) gives \(22x^{-3}\).
• Integrating \(-66x^2\) gives \(-22x^3\).

Thus, the integral is \(22x^{-3} - 22x^3 + C\), which neatly ties back to the original function \(f(x)\). The presence of \(C\) reminds us that there are infinitely many antiderivatives that differ by a constant.

The power rule is a fundamental tool in both differentiation and integration, especially in handling polynomials. It simplifies the process of finding derivatives and integrals by providing a straightforward formula applicable to terms of the form \(ax^n\).

When using the power rule for differentiation, the rule is \(f'(x) = anx^{n-1}\). This means:

• Multiply the term by the exponent \(n\).
• Decrease the exponent by one to find the derivative.

For example, with \(22x^{-3}\) and \(-22x^3\), the power rule transforms them as indicated, yielding the derivatives \(-66x^{-4}\) and \(-66x^2\), respectively.

The power rule for integration is slightly different but equally important: \(\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C\). Here, you:

• Increase the exponent by one.
• Divide by the new exponent to find the integral.

Applying this to \(-66x^{-4}\) and \(-66x^2\) brings us back to \(22x^{-3} - 22x^3 + C\). Understanding the power rule helps simplify complex problems and is a core concept necessary to master calculus.