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The first derivative, often notated as \( r' \), represents the rate of change of a function. It tells us how the function is increasing or decreasing at any given point. To find the first derivative of your function, you'll look at each term separately. This function is composed of fractions, leading us to first rewrite each part with negative exponents. For instance:

• The term \( \frac{12}{\theta} \) becomes \( 12\theta^{-1} \)
• \( \frac{4}{\theta^3} \) changes to \( 4\theta^{-3} \)
• And \( \frac{1}{\theta^4} \) is \( \theta^{-4} \)

Once rewritten, the power rule helps us differentiate each part. The derivative of \( a\theta^{-n} \) turns into \(-na\theta^{-n-1} \). For example, the derivative of \( 12\theta^{-1} \) results in \(-12\theta^{-2} \). Through this approach, calculate each term's derivative and sum them together. The first derivative might also identify critical points, indicating potential local maxima or minima depending on the application.

When you have the first derivative, you might also want to know about the concavity and inflection points of the function, which the second derivative provides. It's represented as \( r'' \) and involves differentiating the first derivative. Simply apply the power rule again to each term of the first derivative.

• From \(-12\theta^{-2} \), you'll derive \( 24\theta^{-3} \)
• From \(-12\theta^{-4} \), to \( 48\theta^{-5} \)
• And from \(-4\theta^{-5} \), proceed to \( 20\theta^{-6} \)

Add together these derivatives to express \( r'' = 24\theta^{-3} + 48\theta^{-5} + 20\theta^{-6} \). This function tells us where the curve is concave upward or downward. Positive values indicate a concave upward shape, while negative values point to concave downward. If the second derivative crosses zero, it can also signal inflection points, where the function changes type of its concavity.

The power rule is an essential tool in calculus for finding derivatives quickly and effectively. It is particularly helpful when dealing with polynomial expressions or terms expressed with negative exponents, as seen in transformations like \( a\theta^{-n} \).
To apply the power rule, follow this easy step:

• Multiply the exponent \(-n\) by the coefficient \(a\), resulting in \(-na \)
• Then, subtract one from the exponent \(-n\) to get the new power: \(-n-1\)

For example, differentiating \( 12\theta^{-1} \), you apply the power rule to get \(-12\theta^{-2} \). This rule is versatile, working for any real number exponent, and simplifies the process of finding how a function changes. By utilizing the power rule, you can streamline the process of solving derivative problems, ultimately saving time and reducing errors.