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Chapter 3: Problem 11

Find the indicated derivatives. $$\frac{d p}{d q} \text { if } \quad p=q^{3 / 2}$$

Short Answer

Expert verified
\( \frac{dp}{dq} = \frac{3}{2} \cdot q^{1/2} \)

Step by step solution

01

Identify the Problem

We are asked to find the derivative \( \frac{dp}{dq} \) where \( p = q^{3/2} \). This is a basic differentiation problem using a power function.
02

Apply the Power Rule

The power rule states that if \( p = q^n \), then \( \frac{dp}{dq} = n \cdot q^{n-1} \). Here, \( n = \frac{3}{2} \).
03

Differentiate the Function

Using the power rule, differentiate \( p = q^{3/2} \). Thus, \( \frac{dp}{dq} = \frac{3}{2} \cdot q^{3/2 - 1} = \frac{3}{2} \cdot q^{1/2} \).
04

Simplify the Derivative

Simplify the expression \( \frac{3}{2} \cdot q^{1/2} \) if necessary. In this case, it remains \( \frac{3}{2} \cdot q^{1/2} \).

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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, especially when working with differentiation. It provides a straightforward method to find the derivative of functions of the form \( q^n \). To apply the power rule, you need to:
  • Identify the exponent \( n \) in the function \( p = q^n \).
  • Multiply the entire term by \( n \).
  • Reduce the power of \( q \) by one, resulting in \( q^{n-1} \).
For example, if \( p = q^{3/2} \), the power rule tells us that the derivative \( \frac{dp}{dq} \) will be \( \frac{3}{2} \cdot q^{3/2 - 1} = \frac{3}{2} \cdot q^{1/2} \). This rule is crucial because it simplifies the process of finding derivatives of polynomial terms or similar expressions found in many calculus problems.
Derivative
A derivative represents how a function changes as its input changes. In simpler terms, it is the rate at which one quantity changes with respect to another. Understanding derivatives is key to understanding change and motion, and they are a cornerstone in calculus.
  • For the function \( p = q^{3/2} \), the derivative \( \frac{dp}{dq} \) tells us how \( p \) changes as \( q \) changes.
  • Calculating the derivative involves the application of rules, like the power rule, to find this rate of change.
  • In real-life scenarios, derivatives help in finding slopes of curves or optimizing processes by understanding trends.
Derivatives also have important physical interpretations such as speed or acceleration in physics, making them extremely practical beyond theoretical exercises.
Basic Differentiation
Basic differentiation is the process of calculating the derivative of a function. This usually involves applying simple rules like the power rule to functions involving only one variable. Learning basic differentiation is essential before moving on to more complex functions or multiple variables.
  • The focus is on identifying simple patterns like power of \( q \) in the function \( p = q^{3/2} \).
  • Once recognized, differentiation turns into a straightforward application of the power and other fundamental rules.
  • It sets the foundation for tackling more advanced topics involving sums, products, or compositions of functions.
The example \( p = q^{3/2} \) displayed a classic case of basic differentiation, emphasizing simplicity and clarity when learning the subject.

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Most popular questions from this chapter

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{25} e^{x}-\log _{5} \sqrt{x}$$

The effect of flight maneuvers on the heart The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation $$ W=P V+\frac{V \delta v^{2}}{2 g} $$ where \(W\) is the work per unit time, \(P\) is the average blood pressure, \(V\) is the volume of blood pumped out during the unit of time, \(\delta\) ("delta") is the weight density of the blood, \(v\) is the average velocity of the exiting blood, and \(g\) is the acceleration of gravity. When \(P, V, \delta,\) and \(v\) remain constant, \(W\) becomes a function of \(g\), and the equation takes the simplified form $$ W=a+\frac{b}{g}(a, b \text { constant }) $$ As a member of NASA's medical team, you want to know how sensitive \(W\) is to apparent changes in \(g\) caused by flight maneuvers, and this depends on the initial value of \(g\). As part of your investigation, you decide to compare the effect on \(W\) of a given change \(d g\) on the moon, where \(g=5.2 \mathrm{ft} / \mathrm{sec}^{2}\), with the effect the same change \(d_{8}\) would have on Earth, where \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\). Use the simplified equation above to find the ratio of \(d W_{\text {moon }}\) to \(d W_{\text {Barth }}\)

Use logarithmic differentiation or the method in Example 7 to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{\ln x}$$

a. Find the linearization of \(f(x)=\log _{3} x\) at \(x=3 .\) Then round its coefficients to two decimal places. b. Graph the linearization and function together in the window \(0 \leq x \leq 8\) and \(2 \leq x \leq 4\)

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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