/*! 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 39 Use the Chain Rule to find the d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 39

Use the Chain Rule to find the derivative of the following functions. $$y=\left(2 x^{6}-3 x^{3}+3\right)^{25}$$

Short Answer

Expert verified
Question: Find the derivative of the function $$y=(2x^6 - 3x^3 + 3)^{25}$$. Answer: The derivative of the function is $$y' = 25(2x^6 - 3x^3 + 3)^{24} \cdot (12x^5 - 9x^2)$$.

Step by step solution

01

Identify the inner and outer functions

The given function can be written as a composite function $$y=(f(g(x)))^{25}$$ where $$f(u) = u^{25}$$ and $$g(x) = 2x^6 - 3x^3 + 3$$ The outer function is \(f(u) = u^{25}\), and the inner function is \(g(x) = 2x^6 - 3x^3 + 3\). Step 2: Find the derivatives of the inner and outer functions
02

Find the derivatives of the inner and outer functions

First, let's find the derivative of the inner function: $$g'(x) = \frac{d}{dx}\left(2x^6 - 3x^3 + 3\right) = 12x^5 - 9x^2$$ Next, find the derivative of the outer function: $$f'(u) = \frac{d}{dx}\left(u^{25}\right) = 25u^{24}$$ Step 3: Apply the Chain Rule
03

Apply the Chain Rule

Now, we apply the chain rule to find the derivative of the composite function: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ where \(u=g(x)\). Substitute the derivatives found in Step 2: $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = 25(2x^6 - 3x^3 + 3)^{24} \cdot (12x^5 - 9x^2)$$ Step 4: Simplify
04

Simplify

Our final answer for the derivative of the function is: $$y' = 25(2x^6 - 3x^3 + 3)^{24} \cdot (12x^5 - 9x^2)$$

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!

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

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=x^{3}+3$$

Quotient Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x\), find a formula for \(\frac{d^{2}}{d x^{2}}\left[\frac{f(x)}{g(x)}\right]\)

Prove the following identities and give the values of \(x\) for which they are true. $$\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}$$

Continuity of a piecewise function Let $$f(x)=\left\\{\begin{aligned} \frac{3 \sin x}{x} & \text { if } x \neq 0 \\ a\ \ \ \ \ & \text { if } x=0 \end{aligned}\right.$$ For what values of \(a\) is \(f\) continuous?

Prove the following identities and give the values of \(x\) for which they are true. $$\cos \left(2 \sin ^{-1} x\right)=1-2 x^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.