/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Differentiate. $$y=(2 x+4)^{3}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 9

Differentiate. $$y=(2 x+4)^{3}$$

Short Answer

Expert verified
\(\frac{dy}{dx} = 6(2x + 4)^2\)

Step by step solution

01

Apply the Chain Rule

To differentiate the function, use the chain rule. The chain rule states that if you have a composite function \(y = f(g(x))\), then the derivative \(y'\) can be found as \(f'(g(x)) \times g'(x)\). Here, \(y=((2x+4)^3)\). Let \(u = 2x + 4\). Then, \(y = u^3\).
02

Differentiate the Outer Function

Differentiate \(y=u^3\) with respect to \(u\). This gives \(\frac{dy}{du}=3u^2\).
03

Differentiate the Inner Function

Differentiate \(u = 2x + 4\) with respect to \(x\). This gives \(\frac{du}{dx}=2\).
04

Combine the Results

Now, use the chain rule to combine the results: \(\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 3u^2 \times 2\).
05

Substitute Back \(u\)

Replace \(u\) with the original expression \(2x + 4\): \(\frac{dy}{dx} = 3(2x + 4)^2 \times 2 = 6(2x + 4)^2\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Chain Rule
The Chain Rule is a fundamental technique in calculus used for finding the derivative of composite functions. Imagine you have a function nested inside another function, like a Russian matryoshka doll.
To make things simpler, think of it as follows: if you have a function defined as \(y = f(g(x))\), then you can take its derivative using the rule: \(y' = f'(g(x)) \times g'(x)\). Basically, you'll first differentiate the outer function, and then multiply it by the derivative of the inner function.
Composite Function
A composite function is like a machine within a machine. The output of one function becomes the input for another. For example, in the function \(y=(2x + 4)^3\), \(2x + 4\) is nested inside the cube function \(u^3\). Let鈥檚 break this down:
  • First, identify the inner function as \(u = 2x + 4\).

  • Then the outer function becomes \(y = u^3\).

By understanding this nesting, you can apply the Chain Rule more effectively.
Differentiation Steps
To differentiate composite functions like \(y = (2x+4)^3\), follow these steps:

Step 1: Apply the Chain Rule
Express the original function in terms of a new variable. Here, we let \(u = 2x + 4\) and rewrite the function as \(y = u^3\).

Step 2: Differentiate the Outer Function
Take the derivative of \(y = u^3\) with respect to \(u\). This gives \(\frac{dy}{du} = 3u^2\).

Step 3: Differentiate the Inner Function
Now, differentiate \(u = 2x + 4\) with respect to \(x\), yielding \(\frac{du}{dx} = 2\).

Step 4: Combine the Results
Use the Chain Rule to get \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 3u^2 \times 2\).

Step 5: Substitute Back \(u\)
Finally, replace \(u\) with \(2x + 4\) to find \(\frac{dy}{dx} = 6(2x+4)^2\).

Following these steps makes differentiation simpler and easier to understand.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(0), \text { where } f(x)=2^{x}$$

Revenue and Marginal Revenue Let \(R(x)\) denote the revenue (in thousands of dollars) generated from the production of \(x\) units of computer chips per day, where each unit consists of 100 chips. (a) Represent the following statement by equations involving \(R\) or \(R^{\prime}:\) When 1200 chips are produced per day, the revenue is 22,000 dollars and the marginal revenue is .75 dollars per chip. (b) If the marginal cost of producing 1200 chips is 1.5 dollars per chip, what is the marginal profit at this production level?

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$

(a) Draw the graph of any function \(f(x)\) that passes through the point (3,2) (b) Choose a point to the right of \(x=3\) on the \(x\) -axis and label it \(3+h\) (c) Draw the straight line through the points \((3, f(3))\) and \((3+h, f(3+h))\) (d) What is the slope of this straight line (in terms of \(h\) )?

In a psychology experiment, people improved their ability to recognize common verbal and semantic information with practice. Their judgment time after \(t\) days of practice was \(f(t)=.36+.77(t-.5)^{-0.36}\) seconds. (Source: American Journal of Psychology.) (a) Display the graphs of \(f(t)\) and \(f^{\prime}(t)\) in the window \([.5,6]\) by \([-3,3]\). Use these graphs to answer the following questions. (b) What was the judgment time after 4 days of practice? (c) After how many days of practice was the judgment time about .8 second? (d) After 4 days of practice, at what rate was judgment time changing with respect to days of practice? (e) After how many days was judgment time changing at the rate of \(-.08\) second per day of practice?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.