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Chapter 3: Problem 40

In Exercises \(31-42,\) find \(d y / d x\). $$y=3\left(2 x^{-1 / 2}+1\right)^{-1 / 3}$$

Short Answer

Expert verified
The derivative of \(y\) with respect to \(x\) is \(dy/dx = x^{-3/2}(2x^{-1/2} + 1)^{-4/3}\)

Step by step solution

01

Identify the outer and inner functions

The outer function is \( f(u) = 3u^{-1/3} \), and the inner function \( u = 2x^{-1/2} + 1 \). The derivative will make use of chain rule which requires differentiating both the inner and outer functions.
02

Differentiate the outer function

Keep \( u = 2x^{-1/2} + 1 \) constant while differentiating. \n For \( f(u) = 3u^{-1/3} \), apply the rule for derivative of \(x^n\), giving \(f'(u) = 3(-1/3)u^{-1/3-1} = -u^{-4/3}\).
03

Differentiate the inner function

Keep the outer function constant while differentiating: \n For \(u = 2x^{-1/2} + 1 \), the derivative is \(u'(x) = 2(-1/2)x^{-1/2-1} = -x^{-3/2}\).
04

Multiply the derivatives using the chain rule

According to chain rule, \(dy/dx = f'(u) \cdot u'(x)\). Substitute the differentiated outer and inner functions from steps 2 and 3 to obtain: \(dy/dx = -u^{-4/3} \cdot -x^{-3/2}\).
05

Substitute the inner function

Substitute \( u = 2x^{-1/2} + 1 \) back into the derivative to obtain the final form of \(dy/dx\).

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Key Concepts

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

Derivative
The derivative is a fundamental concept in calculus, and it measures how a function changes as its input changes. In simpler terms, it indicates the rate of change or slope of the function.
In the context of this exercise, finding the derivative of a function like \( y = 3(2x^{-1/2} + 1)^{-1/3} \) involves several steps.
We use the chain rule, which is especially useful when dealing with composite functions, where one function is "inside" another. This exercise highlights the importance of derivatives and their calculation in understanding the behavior of complex functions. It's similar to peeling layers of an onion, revealing different depths and insights at each layer.
Outer Function
For composite functions, identifying the outer function is crucial. It's the outermost expression that "wraps" other operations inside it.
In this exercise, the outer function is given as \( f(u) = 3u^{-1/3} \). Here, \( u \) represents any expression inside the parentheses.
The role of the outer function is pivotal when applying the chain rule. You maintain the inner function as a constant while differentiating the outer function first. This ensures that you handle the structure of the composite function correctly before diving into the complexity of the inner expressions.
  • Outer functions help set up for differentiating composite expressions efficiently.
  • They guide the first part of differentiation using the chain rule.
Inner Function
The inner function is essentially the expression nested within another function when dealing with composites. It's what you notice after stripping away the outermost layer of a composite expression.
In this context, the inner function is \( u = 2x^{-1/2} + 1 \). The chain rule requires us to handle this expression separately from the outer function.
Analyzing the inner function independently, you gain insight into how changes in \( x \) affect \( u \), which in turn affects the composite function. Keeping this function as a separate entity is key to applying the chain rule accurately.
  • Inner functions require careful differentiation on their own.
  • They provide vital information for completing the chain rule process.
Power Rule
The power rule is a straightforward yet powerful tool in calculus for finding the derivative of expressions in the form of \( x^n \). It states that the derivative of \( x^n \) is \( nx^{n-1} \).
This rule simplifies the process of differentiation, especially when dealing with polynomial expressions.
In our exercise, the power rule is applied when differentiating both the outer and inner functions:
  • For the outer function: \( f(u) = 3u^{-1/3} \), the power rule gives us \( f'(u) = -u^{-4/3} \).
  • For the inner function: \( u = 2x^{-1/2} \), the derivative becomes \( -x^{-3/2} \) using this rule.
The power rule dramatically eases these computations, thus showcasing its significance in calculus. Understanding and using the power rule allows for quicker and more efficient problem-solving.

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Most popular questions from this chapter

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=\log _{10} e^{x}$$

In Exercises \(37-42,\) find \(f^{\prime}(x)\) and state the domain of \(f^{\prime}\) $$f(x)=\ln (2-\cos x)$$

Finding Profit The monthly profit (in thousands of dollars) of a software company is given by \(P(x)=\frac{10}{1+50 \cdot 2^{5-0.1 x}}\) where x is the number of software packages sold. (a) Graph \(P(x)\) (b) What values of \(x\) make sense in the problem situation? (c) Use NDER to graph \(P^{\prime}(x) .\) For what values of \(x\) is \(P\) relatively sensitive to changes in \(x\) ? (d) What is the profit when the marginal profit is greatest? (e) What is the marginal profit when 50 units are sold 100 units, 125 units, 150 units, 175 units, and 300 units? (f) What is \(\lim _{x \rightarrow \infty} P(x) ?\) What is the maximum profit possible? (g) Writing to Learn Is there a practical explanation to the maximum profit answer?

Multiple Choice Which of the following is \(\frac{d}{d x} \sin ^{-1}\left(\frac{x}{2}\right) ?\) \((\mathbf{A})-\frac{2}{\sqrt{4-x^{2}}} \quad(\mathbf{B})-\frac{1}{\sqrt{4-x^{2}}} \quad\) (C) \(\frac{2}{4+x^{2}}\) (D) \(\frac{2}{\sqrt{4-x^{2}}} \quad\) (E) \(\frac{1}{\sqrt{4-x^{2}}}\)

Draining a Tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula \(y=6\left(1-\frac{t}{12}\right)^{2} \mathrm{m}\) (a) Find the rate \(d y / d t(\mathrm{m} / \mathrm{h})\) at which the water level is changing at time. (b) When is the fluid level in the tank falling fastest? slowest? What are the values of \(d y / d t\) at these times? (c) Graph \(y\) and \(d y / d t\) together and discuss the behavior of \(y\) in relation to the signs and values of \(d y / d t .\)
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