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To understand derivatives, let's start with the first derivative, which represents the rate of change or the slope of the curve at any given point. If you imagine walking along a curved path, the rate at which your elevation changes is like the first derivative. For a function like our example, you use the power rule to find this derivative.The function we dealt with is rewritten as \(w = 3z^{-2} - z^{-1}\). The power rule helps us differentiate any term of the form \(z^n\), giving us \(nz^{n-1}\). To apply this:

• For the term \(3z^{-2}\), bring down the exponent \(-2\) to multiply with \(3\), resulting in \(-6z^{-3}\).
• For the term \(-z^{-1}\), bring down the exponent \(-1\) to multiply by \(-1\), resulting in \(z^{-2}\).

So, the first derivative of the function, which shows how it changes, is \(w' = -6z^{-3} + z^{-2}\). This is a vital step in calculus, revealing the function's behavior at any point on the curve.

The second derivative gives us insight into the curvature or concavity of the function, essentially telling us how the slope itself is changing. If the first derivative was about moving up or down a hill, the second derivative tells us whether the hill is getting steeper or less steep.For our example, we started with the first derivative \(w' = -6z^{-3} + z^{-2}\), and we apply differentiation once more:

• The derivative of \(-6z^{-3}\) is \(18z^{-4}\), calculated using the power rule by bringing down the \(-3\) and reducing the exponent.
• The derivative of \(z^{-2}\) is \(-2z^{-3}\), again using the power rule.

Combining these, the second derivative is \(w'' = 18z^{-4} - 2z^{-3}\). This second derivative tells us how the curve bends or twists, indicating points of inflection where the concavity changes.

The power rule is a fundamental tool in calculus used for finding derivatives. It allows us to differentiate any power of a variable quickly and easily. This rule states that for any function \(z^n\), its derivative is \(nz^{n-1}\).This rule simplifies the process of differentiation:

• It works by moving down the exponent as a coefficient in front of the term.
• Then, reduce the original exponent by one.

For instance, in the term \(z^{-2}\), applying the power rule means multiplying \(-2\) by the coefficient, and then lowering the exponent to \(-3\), resulting in \(-2z^{-3}\).By using the power rule, we can quickly determine the derivative of both simple and complex functions with terms involving powers. This makes the power rule a powerful ally in any calculus toolkit, allowing students to tackle derivative problems with confidence and precision.