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The power rule is a fundamental rule for finding derivatives in calculus. It allows us to differentiate expressions where the variable is raised to a power. When using the power rule, if you have a function of the form \( f(z) = az^n \), the derivative is found by multiplying the exponent \( n \) by the coefficient \( a \), and then reducing the exponent by one.
For example, in the term \( 3z^7 \), applying the power rule gives us \( 7 \times 3z^{7-1} = 21z^6 \). This method is efficient for polynomial expressions and forms the backbone of more complex differentiation tasks.
The simplicity of the power rule is due to the fact that each term is treated separately. Ensure to reduce the exponent by exactly one, which reflects how the rate of change is inherently tied to each increase in power.

The first derivative of a function provides us with valuable information about the slope or inclination of the curve at any given point. In simpler terms, it tells us how fast or slow the function is changing.
For the function \( w = 3z^7 - 7z^3 + 21z^2 \), applying the power rule to each term separately gives us the first derivative:

• \( 21z^6 \)
• \(-21z^2 \)
• \(42z \)

Thus, the expression for the first derivative is \( \frac{dw}{dz} = 21z^6 - 21z^2 + 42z \). This expression can tell us the nature of the curve, such as whether it is increasing or decreasing at various points, and where it might have a peak or a dip.

Finding the second derivative involves differentiating the first derivative. While the first derivative tells us about the rate of change, the second derivative gives information on the curvature or concavity of the function.
For the expression \( 21(z^6 - z^2 + 2z) \), applying the power rule once more to each term, you get:

• \( 126z^5 \)
• \(-42z \)
• \(42 \)

This results in the second derivative \( \frac{d^2w}{dz^2} = 126z^5 - 42z + 42 \). The second derivative can indicate the points of inflection, where the curve changes from concave to convex or vice versa.

Differentiation is a key operation in calculus that focuses on computing the derivative of functions. It is vital in understanding the behavior of a function's graph. In practical terms, differentiation helps in finding the slope and curvature, analyzing rates of change, and determining local extrema.
The steps for differentiation generally include:

• Identifying each term in the function
• Applying rules such as the power rule to find derivatives
• Simplifying the expression to get cleaner results

In the exercise, differentiation begins by identifying the function \( w = 3z^7 - 7z^3 + 21z^2 \) and then progressing through finding the first and second derivatives using the power rule. This technique provides a structured approach to analyzing and solving problems involving rates and slopes efficiently.