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The power rule is one of the most straightforward tools in calculus, particularly when dealing with polynomial functions. This rule is applied when you need to find the derivative of a function of the form \(s^n\), where \(n\) is any real number. The rule can be stated as follows: the derivative of \(s^n\) with respect to \(s\) is \(n\times s^{n-1}\).
To apply the power rule successfully, follow these steps:

• Identify each term in the function that is in the form of \(s^n\).
• Multiply the coefficient of \(s^n\) by the exponent \(n\).
• Lower the power of \(s\) by one (from \(n\) to \(n-1\)).

Using the power rule makes the differentiation process efficient and reduces the potential for mistakes.

The first derivative of a function gives you the slope of the tangent line to the function at any given point. In practical terms, it tells you the rate at which the function is changing at that point. For our function \(f(s) = 32s^3 - 2.1s^2 + 7s\), applying the power rule to each term allows you to find the first derivative efficiently.
Here's how the first derivative \(f'(s)\) is obtained:

• For the term \(32s^3\), apply the power rule to get \(96s^2\).
• For the term \(-2.1s^2\), the derivative is \(-4.2s\).
• The linear term \(7s\) becomes \(7\) after differentiation.

Combining these results, the first derivative is \(f'(s) = 96s^2 - 4.2s + 7\). This derivative can be used to understand how quickly the function \(f(s)\) is increasing or decreasing, and can help in finding maximum or minimum points on a graph.

The second derivative provides information about the curvature or the concavity of the original function. It indicates how the rate of change itself is changing. For instance, a positive second derivative means the function is concave up, while a negative second derivative means it is concave down.
To find the second derivative of \(f(s) = 32s^3 - 2.1s^2 + 7s\), you differentiate the first derivative \(f'(s) = 96s^2 - 4.2s + 7\):

• The derivative of \(96s^2\) is \(192s\).
• The derivative of \(-4.2s\) is \(-4.2\).
• The derivative of the constant \(7\) is \(0\).

Putting these terms together, the second derivative is \(f''(s) = 192s - 4.2\). This output helps us understand whether the original function is behaving in an accelerating (positive second derivative) or decelerating (negative second derivative) manner, adding depth to our analysis of function behavior.