/*! 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 2 Differentiate. $$y=3 x^{4}$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 2

Differentiate. $$y=3 x^{4}$$

Short Answer

Expert verified
The derivative is \(12x^3\).

Step by step solution

01

Identify the Power Rule

To differentiate a function of the form \(y = ax^n\), the Power Rule can be used. The Power Rule states that the derivative of \(ax^n\) is \(a \times n \times x^{n-1}\).
02

Apply the Power Rule

In the given function \(y = 3x^4\), identify \(a = 3\) and \(n = 4\). According to the Power Rule, the derivative is found by multiplying the coefficient by the exponent and then subtracting one from the exponent. Thus, \[ \frac{d}{dx}(3x^4) = 3 \times 4 \times x^{4-1} \]
03

Simplify the Expression

Now simplify the expression: \[ 3 \times 4 \times x^{4-1} = 12x^3 \]

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
The Power Rule is a fundamental concept in differentiation. It helps you quickly find the derivative of functions in the form of \[ax^n\]. To use the Power Rule, follow these simple steps:
1. Identify the coefficient (\(a\)) and the exponent (\(n\)).
2. Multiply the coefficient by the exponent.
3. Subtract one from the exponent.
For example, if you have \[y = 3x^4\], the coefficient (\(a\)) is 3, and the exponent (\(n\)) is 4. Applying the Power Rule:
\[ \frac{d}{dx}(3x^4) = 3 \times 4 \times x^{4-1} = 12x^3 \] The final derivative is \[12x^3\]. You just multiplied 3 by 4 and then subtracted 1 from 4 for the new exponent.
Derivatives
A derivative represents the rate at which a function is changing at any given point. It is a measure of how a function's output changes as its input changes. In simpler terms, it's the slope of the function at any point.
When you differentiate a function, you're finding its derivative. For instance, differentiating \[y = 3x^4\] gives us \[ \frac{d}{dx}(3x^4) = 12x^3 \]. This derivative, \[12x^3\], tells us how quickly \[y = 3x^4\] is changing for any given value of \(x\). The steeper the function, the larger the value of the derivative.
Simplification of Expressions
After applying differentiation rules, it's important to simplify the resulting expressions. Simplifying makes the results clearer and more useful.
In our original exercise, we started with differentiating \[y = 3x^4\].
Using the Power Rule, we found:
\[ \frac{d}{dx}(3x^4) = 3 \times 4 \times x^{4-1} \]
Next, simplify by performing the multiplication:
\[ 3 \times 4 = 12 \] and
\[ 4-1 = 3 \].
Thus, the simplified derivative is:
\[ 12x^3 \] Always ensure your final answer is in its simplest form to avoid confusion and make it as clear as possible.

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

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=x^{2}$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x^{3} & \text { for } 0 \leq x<1 \\ x & \text { for } 1 \leq x \leq 2 \end{array}\right.$$

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 7} \frac{x^{3}-2 x^{2}+3 x}{x^{2}}$$

Let \(f(x)\) be the value in dollars of one share of a company \(x\) days since the company went public. (a) Interpret the statements \(f(100)=16\) and \(f^{\prime}(100)=.25.\) (b) Estimate the value of one share on the 101 st day since the company went public.

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x+2 & \text { for }-1 \leq x \leq 1 \\ 3 x & \text { for } 1
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.