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Chapter 3: Problem 7

Find the first and second derivatives. $$ w=3 z^{-2}-\frac{1}{z} $$

Short Answer

Expert verified
The first derivative is \( -6z^{-3} + z^{-2} \) and the second derivative is \( 18z^{-4} - 2z^{-3} \).

Step by step solution

01

Rewrite the Function

First, let's rewrite the function more conveniently for differentiation. The given function is \( w = 3z^{-2} - \frac{1}{z} \). Rewrite \( \frac{1}{z} \) as \( z^{-1} \). Now the function becomes \( w = 3z^{-2} - z^{-1} \).
02

Find the First Derivative

To find the first derivative \( \frac{dw}{dz} \), apply the power rule to each term. The power rule states that the derivative of \( z^n \) is \( nz^{n-1} \). Starting with \( 3z^{-2} \), the derivative is \( 3 \times (-2)z^{-3} = -6z^{-3} \). For \( -z^{-1} \), the derivative is \( -1 \times (-1)z^{-2} = z^{-2} \). Combining these, the first derivative is \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \).
03

Find the Second Derivative

Now, find the second derivative \( \frac{d^2w}{dz^2} \) by differentiating the first derivative. Start with \( -6z^{-3} \); the derivative is \( -6 \times (-3)z^{-4} = 18z^{-4} \). Next, for \( z^{-2} \), the derivative is \( -2z^{-3} \). Thus, the second derivative is \( \frac{d^2w}{dz^2} = 18z^{-4} - 2z^{-3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The Power Rule is a fundamental concept in calculus and is used for finding the derivative of polynomials. It allows us to simplify the process of differentiation significantly.

The rule states that if you have a function in the form of \( f(z) = z^n \), then the derivative of this function with respect to \( z \) is \( f'(z) = nz^{n-1} \). This means you multiply the exponent \( n \) by the coefficient and reduce the exponent by one.

Here's how it works step by step:
  • Identify the term you want to differentiate.
  • Multiply the exponent by the coefficient of the term.
  • Subtract one from the exponent.

For example, in the function \( w = 3z^{-2} \), applying the Power Rule gives us \( -6z^{-3} \) as the derivative.
First Derivative
The first derivative of a function represents the rate of change of the function with respect to its variable. It's often used to determine the slope of the tangent line at any given point on a function's graph. In other words, it provides information about how the function is varying.

In our exercise, to find the first derivative of \( w = 3z^{-2} - z^{-1} \), we apply the Power Rule to each term individually:
  • For \( 3z^{-2} \), the derivative is \( -6z^{-3} \).
  • For \( -z^{-1} \), the derivative is \( z^{-2} \).

Combining these results, the first derivative \( \frac{dw}{dz} \) is: \( -6z^{-3} + z^{-2} \). This tells us the rate at which \( w \) changes with \( z \).
Second Derivative
The second derivative provides information about the curvature or concavity of the function's graph. It shows how the rate of change (first derivative) of the function itself is changing. If the second derivative is positive, the function is concave up; if negative, the function is concave down.

To find the second derivative, you differentiate the first derivative. In our example, the first derivative is \( -6z^{-3} + z^{-2} \). Let's apply the Power Rule again:
  • The derivative of \( -6z^{-3} \) is \( 18z^{-4} \).
  • The derivative of \( z^{-2} \) is \( -2z^{-3} \).

Combining these, the second derivative \( \frac{d^2w}{dz^2} \) becomes: \( 18z^{-4} - 2z^{-3} \). This tells us about the concavity and acceleration of changes in \( w \) related to \( z \).

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Most popular questions from this chapter

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