/*! 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 Derivatives of functions with ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 39

Derivatives of functions with rational exponents Find \(\frac{d y}{d x}\). $$y=(5 x+1)^{2 / 3}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = (5x + 1)^{\frac{2}{3}}\). Answer: The derivative of the function is \(\frac{dy}{dx} = \frac{10}{3}(5x + 1)^{-\frac{1}{3}}\).

Step by step solution

01

Identify the outer and inner functions

The given function is \(y = (5x + 1)^{\frac{2}{3}}\). Here, the outer function is \(F(u) = u^{\frac{2}{3}}\) and the inner function is \(u = G(x) = 5x + 1\). Now we will find the derivative of F(u) and G(x) i.e., F'(u) and G'(x).
02

Differentiate outer function with respect to its variable

Now we need to differentiate the outer function, F(u) = \(u^{\frac{2}{3}}\), with respect to u: $$ F'(u) =\frac{d}{du} (u^{\frac{2}{3}}) $$$$ = \frac{2}{3}u^{\frac{2}{3}-1} $$$$ = \frac{2}{3}u^{-\frac{1}{3}} $$
03

Differentiate inner function with respect to its variable

Now, we need to differentiate the inner function, G(x) = 5x + 1, with respect to x: $$ G'(x) = \frac{d}{dx} (5x + 1) $$$$ = 5 $$
04

Apply the chain rule

The chain rule states that \(y' = F'(G(x)) \cdot G'(x)\). We already found \(F'(u)\) and \(G'(x)\) in step 2 and step 3, now let's apply the chain rule: $$ \frac{dy}{dx} = F'(G(x)) \cdot G'(x) $$$$ = \frac{2}{3}(5x + 1)^{-\frac{1}{3}} \cdot 5 $$
05

Simplify the result

Now, let's simplify our final expression for the derivative: $$ \frac{dy}{dx} = \frac{10}{3}(5x + 1)^{-\frac{1}{3}} $$ So the derivative of the given function is \(\frac{dy}{dx} = \frac{10}{3}(5x + 1)^{-\frac{1}{3}}\).

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.

Rational Exponents
Rational exponents are a key concept in calculus and algebra. They allow us to express roots and powers in a unified way using fractions as exponents. For instance, when we have a function like \(y = (5x + 1)^{\frac{2}{3}}\), the exponent \(\frac{2}{3}\) is rational. This means the expression can be interpreted as both a power and a root.
  • The numerator of the exponent, 2, indicates that the expression is squared.
  • The denominator, 3, indicates that the cubic root is taken.
Thus, \((5x + 1)^{\frac{2}{3}}\) can be written as \(\sqrt[3]{(5x + 1)^2}\). Understanding these equivalents make it easier when differentiating, as they simplify many algebraic expressions during the process of differentiation.
Chain Rule
The chain rule is a fundamental differentiation technique used when dealing with composite functions. A composite function is one where a function is applied to another function, like in our exercise where \(y = (5x + 1)^{\frac{2}{3}}\). Here, we have:
  • The outer function: \(F(u) = u^{\frac{2}{3}}\).
  • The inner function: \(u = 5x + 1\).
To apply the chain rule, differentiate the outer function with respect to its inner function, and then multiply it by the derivative of the inner function. Mathematically, this is:\[F'(G(x)) \cdot G'(x)\]
So, when working on composite functions, identifying these layers and correctly applying the chain rule allows us to find derivatives even for more complicated situations.
Differentiation Techniques
Differentiation techniques are essential tools in calculus. They help us determine how a function changes at any given point. The exercise demonstrates a few key techniques and concepts:
  • Identify Functions: Recognizing inner and outer functions helps when applying rules like the chain rule.
  • Manual Differentiation: Understand the expressions step-by-step. For example, after identifying \(F(u)\) and \(G(x)\), differentiate them independently. For \(F(u)\), using the power rule gives \(\frac{2}{3}u^{-\frac{1}{3}}\).
  • Simplification: After applying the chain rule, simplify the resulting expression. This simplification involves combining constants and reducing expression complexity to reach an easier-to-understand form like: \(\frac{10}{3}(5x + 1)^{-\frac{1}{3}}\).
Mastering these techniques not only aids in finding derivatives efficiently but also develops a deeper understanding of function behaviors.

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

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}=\frac{x^{2}(4-x)}{4+x} \text { (right strophoid) }$$

Use any method to evaluate the derivative of the following functions. $$f(x)=4 x^{2}-\frac{2 x}{5 x+1}$$

Witch of Agnesi Let \(y\left(x^{2}+4\right)=8\) (see figure). a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when \(y=1\) c. Solve the equation \(y\left(x^{2}+4\right)=8\) for \(y\) to find an explicit expression for \(y\) and then calculate \(\frac{d y}{d x}\) d. Verify that the results of parts (a) and (c) are consistent.

A port and a radar station are 2 mi apart on a straight shore running east and west. A ship leaves the port at noon traveling northeast at a rate of \(15 \mathrm{mi} / \mathrm{hr}\). If the ship maintains its speed and course, what is the rate of change of the tracking angle \(\theta\) between the shore and the line between the radar station and the ship at 12: 30 p.m.? (Hint: Use the Law of sines.)

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{f(x)}{(x+2)}\right]\right|_{x=4}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.