91Ó°ÊÓ

When we talk about the first derivative of a function, we are referring to the rate at which the function's value changes as the input changes. In the context of our original exercise, you start by simplifying the given expression for the function \( w = \left(\frac{1+3z}{3z}\right)(3-z) \).

After expanding and simplifying, we arrived at the expression \( w = \frac{1}{z} + \frac{8}{3} - z \). The task is then to find the first derivative, \( w' \), of this simplified function in terms of \( z \).

To find the first derivative, apply the power rule, which states that the derivative of \( z^n \) is \( nz^{n-1} \). Let's see how this applies to the components:

• For \( \frac{1}{z} \), rewrite it as \( z^{-1} \). The derivative becomes \( -z^{-2} \).
• For the constant \( \frac{8}{3} \), the derivative is 0 since constants do not change and hence their derivative is zero.
• For \( -z \), the derivative is simply \(-1\) since the exponent of \( z \) is 1.

Putting it all together, the first derivative is \( w' = -z^{-2} - 1 \). This tells us the slope of the tangent line to the curve of the function at any point \( z \).

The second derivative is a step further in understanding the behavior of a function's graph. It tells us about the curvature of the function, essentially providing insight into the concavity or convexity of the function's graph.

After finding the first derivative \( w' = -z^{-2} - 1 \), we need to find the second derivative, denoted as \( w'' \). Start by differentiating the first derivative.

Using the power rule again, the process is as follows:

• For the term \(-z^{-2}\), apply the power rule. The derivative is \( 2z^{-3} \) because differentiating \( z^{-2} \) results in \(-2z^{-3}\) and considering the negative leads to \( 2z^{-3} \).
• The constant term \(-1\) has a derivative of 0.

Thus, the second derivative is \( w'' = 2z^{-3} \). This shows how the rate of change of the slope of the tangent is affected by changes in \( z \), indicating whether the function is curving up or down at any point.

The power rule is a central tool in calculus for finding derivatives. It's especially useful when dealing with polynomials or any expression that can be rewritten as a sum of powers of the variable.

The rule is simple yet powerful: if you have a term \( z^n \), its derivative is \( nz^{n-1} \). This rule is what you employ to find derivatives in the exercise.

Here’s how it was utilized:

• When finding the derivative of the term \( \frac{1}{z} \), rewrite it as \( z^{-1} \), then apply the power rule to obtain the derivative \(-z^{-2}\).
• For \(-z\), which is \(-z^1\), applying the power rule gives \(-1\) because \(1 \cdot z^{0} = 1\) and \(–1 \cdot 1 = -1\).
• While constants like \( \frac{8}{3} \) are unaffected by changes and have a derivative of 0.

Mastering the power rule allows you to quickly and efficiently handle derivatives, especially in polynomial functions or functions expressed in terms of powers.