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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: \(w' = -6z^{-3} + z^{-2}\). Second derivative: \(w'' = 18z^{-4} - 2z^{-3}\).

Step by step solution

01

Rewrite the Function

First, let's rewrite the function using exponents for easier differentiation: \[ w = 3z^{-2} - z^{-1} \] This expression will allow us to use the power rule more easily.
02

Find the First Derivative

Use the power rule for differentiation, \( \frac{d}{dz}(z^n) = nz^{n-1} \), to find the derivative of each term:- The derivative of \(3z^{-2}\) is \[ \frac{d}{dz}(3z^{-2}) = 3 \cdot (-2)z^{-3} = -6z^{-3} \]- The derivative of \(z^{-1}\) is \[ \frac{d}{dz}(-z^{-1}) = -1 \cdot (-1)z^{-2} = z^{-2} \]Thus, the first derivative, \(w'\), is \[ w' = -6z^{-3} + z^{-2} \]
03

Find the Second Derivative

Differentiate the first derivative to find the second derivative:- The derivative of \(-6z^{-3}\) is \[ \frac{d}{dz}(-6z^{-3}) = -6 \cdot (-3)z^{-4} = 18z^{-4} \]- The derivative of \(z^{-2}\) is \[ \frac{d}{dz}(z^{-2}) = -2z^{-3} \]Thus, the second derivative, \(w''\), is \[ w'' = 18z^{-4} - 2z^{-3} \]

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Key Concepts

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

Power Rule
Understanding the power rule is key for quickly finding derivatives. The power rule states that if you have a function of the form \( f(z) = z^n \), the derivative \( f'(z) \) is \( nz^{n-1} \). This means you multiply the entire term by the exponent, then decrease the exponent by one.
  • For example, if \( f(z) = z^3 \), the derivative is \( f'(z) = 3z^2 \).
  • The term \( 3z^{-2} \) uses the power rule by multiplying \( -2 \) with \( 3 \) to get \( -6 \), and reducing the exponent by 1 to become \(-3 \) for \( -6z^{-3} \).
The power rule simplifies differentiation significantly, especially when dealing with polynomials or any function that can be expressed as powers of a variable.
Higher Order Derivatives
When you find the derivative of a derivative, you are finding a higher order derivative. The first derivative tells you about the rate of change, or slope, of the original function. The second derivative provides information about how this rate of change itself changes.
  • The first derivative \( w' \) was found to be \(-6z^{-3} + z^{-2} \). This describes how the function \( w \) changes with respect to \( z \).
  • The second derivative \( w'' \) is \( 18z^{-4} - 2z^{-3} \). This indicates whether the original function is curving upwards or downwards, thus giving insights into the concavity.
Higher order derivatives are useful in many areas such as physics and engineering, where understanding how systems change is crucial.
Differentiation Techniques
There are several techniques to master when differentiating functions, and understanding when to use each is crucial. In this exercise, we primarily focused on the power rule. However, recognizing the form of the function can help decide which technique to apply.
  • For rational expressions, rewriting them using negative exponents, as done with \( z^{-1} \) and \( z^{-2} \), can facilitate the differentiation process through the power rule.
  • Other common techniques include the product rule, chain rule, and quotient rule for more complex functions not easily rewritten into polynomial form.
Each technique serves as a tool to handle different differentiation scenarios efficiently, enabling a systematic approach to finding derivatives.

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Most popular questions from this chapter

Measuring acceleration of gravity When the length \(L\) of a clock pendulum is held constant by controlling its temperature, the pendulum's period \(T\) depends on the acceleration of gravity \(g\). The period will therefore vary slightly as the clock is moved from place to place on Earth's surface, depending on the change in \(g\). By keeping track of \(\Delta T\), we can estimate the variation in \(g\) from the equation \(T=2 \pi(L / g)^{1 / 2}\) that relates \(T, g,\) and \(L\) a. With \(L\) held constant and \(g\) as the independent variable, calculate \(d T\) and use it to answer parts (b) and (c). b. If \(g\) increases, will \(T\) increase or decrease? Will a pendulum clock speed up or slow down? Explain. c. A clock with a \(100-\mathrm{cm}\) pendulum is moved from a location where \(g=980 \mathrm{cm} / \mathrm{sec}^{2}\) to a new location. This increases the period by \(d T=0.001\) sec. Find \(d g\) and estimate the value of g at the new location.

Estimate the allowable percentage error in measuring the diameter \(D\) of a sphere if the volume is to be calculated correctly to within \(3 \%\)

The derivative of \(\sin 2 x \quad\) Graph the function \(y=2 \cos 2 x\) for \(-2 \leq x \leq 3.5 .\) Then, on the same screen, graph $$ y=\frac{\sin 2(x+h)-\sin 2 x}{h} $$ for \(h=1.0,0.5,\) and \(0.2 .\) Experiment with other values of \(h\) including negative values. What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find an equation for the tangent line to \(f\) at the specified $$ \text { point }\left(x_{0}, f\left(x_{0}\right)\right) $$ d. Find an equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right) .\) Discuss the symmetries you see across the main diagonal (the line \(y=x\) ). $$y=\sqrt{3 x-2}, \quad \frac{2}{3} \leq x \leq 4, \quad x_{0}=3$$

a. Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with these properties: i. \(Q(a)=f(a)\) ii. \(Q^{\prime}(a)=f^{\prime}(a)\) iii. \(Q^{\gamma}(a)=f^{\prime \prime}(a)\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) b. Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0\) c. Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point (0,1) Comment on what you see. d. Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) Graph \(g\) and its quadratic approximation together. Comment on what you see. e. Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. I. What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts (b), (d), and (e)?
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