/*! 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 32 Find each derivative. $$\frac{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 32

Find each derivative. $$\frac{d}{d x}\left(6 x^{2}-5 x+9\right)$$

Short Answer

Expert verified
\[ 12x - 5 \].

Step by step solution

01

Identify the Function

The function to differentiate is given as \[ f(x) = 6x^2 - 5x + 9 \].
02

Apply the Power Rule to the First Term

The first term is \[ 6x^2 \]. Using the power rule, which states \[ \frac{d}{dx}(x^n) = nx^{n-1} \], we get \[ \frac{d}{dx}(6x^2) = 6 \times 2x^{2-1} = 12x \].
03

Apply the Power Rule to the Second Term

The second term is \[ -5x \]. Using the power rule, we get \[ \frac{d}{dx}(-5x) = -5 \times 1x^{1-1} = -5 \].
04

Apply the Power Rule to the Constant Term

The third term is \[ 9 \], which is a constant. The derivative of a constant is 0, so \[ \frac{d}{dx}(9) = 0 \].
05

Combine the Results

Now combine the derivatives of each term: \[ 12x - 5 + 0 = 12x - 5 \].

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
The power rule is a fundamental tool in differentiation. It's used to find the derivative of a term that has a variable raised to an exponent. The rule states: \ \( \frac{d}{dx}(x^n) = nx^{n-1} \)
This means you multiply the exponent by the coefficient and then reduce the exponent by one.
For example, if you have a function term like \ 6x^2 \, applying the power rule works as follows: \ \( \frac{d}{dx}(6x^2) = 6 \times 2x^{2-1} = 12x \)
Remember, every time you see \( x^n \), you can use this method to find its derivative. It's a quick way to handle polynomial functions!
constant term differentiation
Constant terms are the simplest part of a function to differentiate. A constant term is a number without any variables attached.
The rule for differentiating a constant is straightforward: the derivative of a constant is always zero.
For instance, if you have a constant term like 9, its derivative is \( \frac{d}{dx}(9) = 0 \).
This is because constants do not change as the variable changes. Since differentiation measures how a function changes, constants have no change, and thus their derivative is zero.
combining derivatives
Once you have found the derivatives of each term in your function, the final step is to combine them.
Let's say you have calculated the following derivatives:
  • \(12x \) from the term \( 6x^2 \),
  • \( -5 \) from the term \( -5x \),
  • \ 0 \ from the constant term 9.

You simply add these results together to get your final derivative: \ 12x - 5 + 0 = 12x - 5 \.
By following this method, you can efficiently combine the derivatives of all the terms to arrive at the solution.

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

Business profit. A company is selling laptop computers. It determines that its total profit, in dollars, is given by \(P(x)=0.08 x^{2}+80 x\) where \(x\) is the number of units produced and sold. Suppose that \(x\) is a function of time, in months, where \(x=5 t+1\) a) Find the total profit as a function of time \(t\) b) Find the rate of change of total profit when \(t=48\) months.

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=(x+3)(x-2)$$

Business and Economics A total-revenue function is given by \(R(x)=1000 \sqrt{x^{2}-0.1 x}\) where \(R(x)\) is the total revenue, in thousands of dollars, from the sale of \(x\) items. Find the rate at which total revenue is changing when 20 items have been sold.

Differentiate each function. $$f(t)=\left(t^{5}+3\right) \cdot \frac{t^{3}-1}{t^{3}+1}$$

Differentiate each function. $$f(x)=\frac{3 x^{2}+2 x}{x^{2}+1}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.