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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
First derivative: \( -6z^{-3} + z^{-2} \); Second derivative: \( 18z^{-4} - 2z^{-3} \).

Step by step solution

01

Rewrite the function for easier differentiation

We have the function \( w = 3z^{-2} - \frac{1}{z} \). Rewriting \( \frac{1}{z} \) in exponential form gives \( z^{-1} \). Now the function is \( w = 3z^{-2} - z^{-1} \).
02

Differentiate to find the first derivative

Use the power rule \( \frac{d}{dz}z^n = nz^{n-1} \) to differentiate. For \( 3z^{-2} \), the derivative is \( -6z^{-3} \), and for \( -z^{-1} \), the derivative is \( z^{-2} \). The first derivative is \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \).
03

Differentiate again to find the second derivative

Differentiate the first derivative \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \). For \( -6z^{-3} \), the derivative is \( 18z^{-4} \), and for \( z^{-2} \), the derivative is \( -2z^{-3} \). 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.

Understanding Derivatives
A derivative is a key concept in calculus, often thought of as a tool that measures how a function changes as its input changes. It provides the rate at which one quantity changes relative to another. When you have a function, say with variable \( z \), its derivative tells you the slope of the tangent to the function at any point on its graph.
  • The first derivative of a function is particularly useful because it describes the function's rate of change or its velocity.
  • This can indicate where the function is increasing or decreasing.
  • For the given function \( w = 3z^{-2} - z^{-1} \), the first derivative \( \frac{dw}{dz} \) reveals the function's behavior with respect to \( z \).
To compute derivatives, we follow a set of differentiation rules, and one commonly used rule is the power rule.
The Power Rule
The power rule is a straightforward way to find derivatives of functions with terms in the form \( z^n \). This rule states:
For any function \( z^n \), \( \frac{d}{dz}z^n = nz^{n-1} \).
It's a fast and effective method that allows us to tackle polynomial functions and power functions with ease.
  • In our example, we applied the power rule to each term individually.
  • For \( 3z^{-2} \), we multiply \( -2 \) by \( 3 \), giving \( -6z^{-3} \).
  • Similarly, for \( -z^{-1} \), multiplying by \( -1 \) results in \( z^{-2} \).
This application of the power rule gives the first derivative: \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \).
Exploring the Second Derivative
The second derivative provides even more insights into a function's behavior. It is the derivative of the first derivative, giving us the rate of change of the rate of change. This can help identify points of inflection and the concavity of the graph.
  • If the second derivative is positive, the function is concave up, resembling a cup \( \cup \).
  • If it's negative, the function is concave down, resembling a cap \( \cap \).
  • In our exercise, we take the first derivative \( -6z^{-3} + z^{-2} \) and differentiate each term again.
Applying the power rule once more:
  • The derivative of \( -6z^{-3} \) is \( 18z^{-4} \).
  • The derivative of \( z^{-2} \) is \( -2z^{-3} \).
Thus, the second derivative is \( \frac{d^2w}{dz^2} = 18z^{-4} - 2z^{-3} \). This helps further describe the curvature of the graph of the function.

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Most popular questions from this chapter

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{5} e^{x}$$

Use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval \(I .\) Perform the following steps: a. Plot the function \(f\) over \(I\) b. Find the linearization \(L\) of the function at the point \(a\). c. Plot \(f\) and \(L\) together on a single graph. d. Plot the absolute error \(|f(x)-L(x)|\) over \(I\) and find its maximum value. e. From your graph in part (d), estimate as large a \(\delta>0\) as you can, satisfying $$|x-a|<\delta \Rightarrow|f(x)-L(x)|<\epsilon$$ for \(\epsilon=0.5,0.1,\) and \(0.01 .\) Then check graphically to see if your \(\delta\) -estimate holds true. $$f(x)=x 2^{x}, \quad[0,2], \quad a=1$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sin x)^{x}$$

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\cot ^{-1}(-1 / 2)\) b. \(\cos ^{-1}(-5)\)

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$$
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