/*! 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 6 Find the derivative of the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 6

Find the derivative of the function. \(f(x)=9 x^{1 / 3}\)

Short Answer

Expert verified
The derivative of the function \(f(x) = 9 x^{\frac{1}{3}}\) is \(f'(x) = 3 x^{-\frac{2}{3}}\).

Step by step solution

01

Identify constants, a and n, in the given function

In our function \(f(x) = 9 x^{\frac{1}{3}}\), the constant \(a = 9\) and the constant \(n = \frac{1}{3}\).
02

Apply the power rule of derivatives

Using the power rule of derivatives, \(f'(x) = anx^{n - 1}\), we'll substitute our identified constants, \(a = 9\) and \(n = \frac{1}{3}\), into the formula.
03

Calculate the derivative

Now that we have our constants, we can calculate the derivative of the function: \(f'(x) = 9 \cdot \frac{1}{3} x^{\frac{1}{3} - 1}\)
04

Simplify the expression

To simplify the expression, we'll perform the arithmetic and simplify the exponent: \(f'(x) = 3 x^{\frac{1}{3} - \frac{3}{3}} = 3 x^{-\frac{2}{3}}\)
05

Write the final answer

The derivative of the function \(f(x) = 9 x^{\frac{1}{3}}\) is: \(f'(x) = 3 x^{-\frac{2}{3}}\)

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.

Power Rule of Derivatives
Understanding the power rule is vital for anyone diving into the world of calculus. Simply put, the power rule is a shortcut that allows us to quickly find the derivative of a function where the variable's exponent is a real number. This rule states that if you have a function of the form f(x) = ax^n, where a is a constant and n is a real number, the derivative of f(x) with respect to x is f'(x) = anx^{n-1}.

For example, let's consider a function like f(x) = 9x^(1/3). Here a = 9 and n = 1/3. By applying the power rule, we derive that f'(x) = 9 * (1/3)x^(1/3 - 1). This makes calculations quick and efficient, bypassing the need for the long process of limits that defines the derivative.
Derivative Calculation
Once you're familiar with various derivative rules like the power rule, derivative calculation becomes a clear process of applying these rules correctly. Let's continue with our example of f(x) = 9x^(1/3). After applying the power rule, we get f'(x) = 9 * (1/3)x^(1/3 - 1). The next step is to perform the operations involved. This means multiplying the coefficients and subtracting one from the exponent.

In our case, 9 * (1/3) simplifies to 3, and (1/3) - 1 simplifies to -2/3 when the exponents are dealt with. Thus, we arrive at f'(x) = 3x^(-2/3). Having a systematic approach to calculate derivatives can greatly improve both understanding and accuracy.
Simplifying Expressions
Simplification is about making expressions more readable and manageable. It is important not to skip this step, as it often makes further algebraic manipulation and interpretation more intuitive. After calculating the derivative, you might end up with an expression that seems complex, like f'(x) = 3x^(-2/3).

This can be further simplified to a more familiar form by rewriting the negative exponent as a reciprocal: f'(x) = 3/x^(2/3). This notation indicates that the derivative involves a division by the variable x raised to the power of 2/3. Simplifying the expressions within derivative calculations is not just about aesthetics; it's about clarity and ease of communication in mathematical expressions.

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

In Exercises, (a) find the equations of the tangent and the normal lines to the curve at the indicated point. (The normal line at a point on the curve is the line perpendicular to the tangent line at that point.) (b) Then use a graphing utility to plot the curve and the tangent and normal lines on the same screen. $$ 4 x y-9=0 ; \quad\left(3, \frac{3}{4}\right) $$

Find the derivative of the function. $$ g(t)=t \tan ^{-1} 3 t $$

Find the derivative of the function. $$ f(x)=\cos ^{-1}(\sin 2 x) $$

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. In Exercises \(89-92\), show that the curves with the given equations are orthogonal.$$ y-x=\frac{\pi}{2}, \quad x=\cos y $$

Let \(g\) denote the inverse of the function \(f\). (a) Show that the point \((a, b)\) lies on the graph of \(f .\) (b) Find \(g^{\prime}(b)\) $$ f(x)=2 x+1 ; \quad(2,5) $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.