/*! 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 1 Differentiate. $$y=6 x^{3}$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 1

Differentiate. $$y=6 x^{3}$$

Short Answer

Expert verified
The derivative is

Step by step solution

01

Identify the function and the differentiation rule

The function we need to differentiate is given by: . Recognize that this function is a power function of the form , where . We will use the power rule for differentiation.
02

Power Rule for Differentiation

The power rule states that if , then . Apply this rule to differentiate the given function.
03

Apply the Power Rule

Using the power rule, differentiate the function with respect to : . Here, .
04

Simplify the Result

Simplify the resulting expression:

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 calculus for finding the derivative of a power function. This rule states that if you have a function in the form \(f(x) = ax^n\), where \(a\) is a constant and \(n\) is an exponent, the derivative of this function is given by \(f'(x) = anx^{n-1}\).
Applicability includes any function where a variable is raised to a power.
  • Identify the constants and power in the function.
  • Bring down the exponent as a multiplier.
  • Subtract one from the original exponent.
For example, in the function \(y = 6x^3\), the constant is 6 and the exponent \(n\) is 3. The derivative, using the power rule, simplifies the process greatly.
Calculus
Calculus is the branch of mathematics that studies continuous change. It's split into two main parts: differential calculus, which deals with rates of change (derivatives), and integral calculus, which focuses on the accumulation of quantities (integrals).
Differentiation falls under differential calculus, helping us determine how a function changes at any given point.
Understanding calculus principles is critical for fields like physics, engineering, economics, and many more.
  • Differential calculus finds how a quantity changes.
  • Integral calculus finds the total amount where the quantity accumulates.
Both parts of calculus work together to solve complex problems related to change and accumulation in various scientific and mathematical contexts.
Derivatives
A derivative represents the rate at which a function is changing at any given point.
It measures the slope of the tangent line to the function's graph at a specific point. The derivative of a function \(f(x)\) is usually denoted as \(f'(x)\) or \(\frac{df}{dx}\).
  • To find the derivative of a function, you use differentiation rules like the power rule.
  • Derivatives have practical applications such as calculating velocities and accelerations, optimizing functions, and modeling real-world problems.
For instance, differentiating \(y = 6x^3\) using the power rule results in \(\frac{dy}{dx} = 18x^2\). This tells us how quickly \(y\) changes for changes in \(x\).
Mastering derivatives and differentiation techniques is essential for solving problems involving dynamic systems.

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

Examine the graph of the function and evaluate the function-atlarge values of \(x\) to guess the value of the limit. $$\lim _{x \rightarrow \infty} \frac{-8 x^{2}+1}{x^{2}+1}$$

If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x.\) $$f(x)=\frac{\sqrt{9+x}-\sqrt{9}}{x}, x \neq 0$$

Determine the value of \(a\) that makes the function \(f(x)\) continuous at \(x=0.\) $$f(x)=\left\\{\begin{array}{ll} 1 & \text { for } x \geq 0 \\ x+a & \text { for } x<0 \end{array}\right.$$

Estimate how much the function $$f(x)=\frac{1}{1+x^{2}}$$ will change if \(x\) decreases from 1 to .9.

If \(s=7 x^{2} y \sqrt{z},\) find: $$(a) \frac{d^{2} s}{d x^{2}} \quad \quad (b) \frac{d^{2} s}{d y^{2}} \quad \quad (c) \frac{d s}{d z}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

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.