91Ó°ÊÓ

Differentiation is a fundamental concept in calculus used to find how a function changes at any given point. A crucial tool to simplify this process is the power rule. The power rule is specifically helpful when dealing with functions where the variable is raised to a power, in the form of \( s^n \).
To apply the power rule, you multiply the function by the exponent \( n \) and then reduce the exponent by one. Mathematically, for a function \( s^n \), its derivative is \( n \times s^{n-1} \).
This rule significantly simplifies finding first and second derivatives, especially when the function is a polynomial or contains roots, such as in the expression \( s^{3/2} \). The power rule supports rapid differentiation, transforming a complex expression into an actionable derivative with less effort.

The first derivative of a function reflects the rate at which the function value changes with respect to changes in its variable. In simpler terms, it measures the slope or steepness of the function graph at any point.
For the function \( f(s) = s^{3/2} \), the first derivative can be calculated using the power rule. By multiplying the power \( \frac{3}{2} \) with the function and reducing the power by one, we get: \[ f'(s) = \frac{3}{2}s^{1/2} \]
This derivative \( f'(s) \) provides important insights, like determining where the function is increasing or decreasing. In our example:

• The derivative value increases when \( s \) increases, suggesting a rising slope.
• At \( s = 0 \), the derivative becomes undefined, which highlights that the slope may tend towards infinity or discontinuity.

Once the first derivative is obtained, it becomes possible to derive further insights by calculating the second derivative. This involves differentiating the first derivative, and it tells us about the concavity or the acceleration of the function.

From the first derivative \( f'(s) = \frac{3}{2}s^{1/2} \), applying the power rule again gives us:
\[ f''(s) = \frac{3}{4}s^{-1/2} \]
The second derivative provides valuable information about the behavior of the function:

• If \( f''(s) > 0 \), the graph is concave upwards, leading to a smile-shaped curve.
• If \( f''(s) < 0 \), the graph is concave downwards, forming a frown.

For \( f(s) = s^{3/2} \), the negative exponent implies that \( f''(s) \) evaluates the rate of change in slope to decrease as \( s \) increases.