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Chapter 2: Problem 3

Find the derivative of the given function. $$ f(x)=6 x^{4}-7 x^{3}+2 x+\sqrt{2} $$

Short Answer

Expert verified
f'(x) = 24x^3 - 21x^2 + 2

Step by step solution

01

Identify the function components

The given function is \[ f(x) = 6x^4 - 7x^3 + 2x + \sqrt{2} \]. Identify each term separately: 1. \(6x^4\)2. \(-7x^3\)3. \(2x\)4. \(\sqrt{2}\)
02

Apply the power rule to polynomial terms

To differentiate each polynomial term, apply the power rule \( \frac{d}{dx} [x^n] = nx^{n-1} \).For \(6x^4\): \[ \frac{d}{dx} [6x^4] = 6 \cdot 4x^{4-1} = 24x^3 \]. For \(-7x^3\): \[ \frac{d}{dx} [-7x^3] = -7 \cdot 3x^{3-1} = -21x^2 \]. For \(2x\): \[ \frac{d}{dx} [2x] = 2 \cdot 1x^{1-1} = 2 \].
03

Differentiate the constant term

The derivative of a constant is always zero. Thus, \[ \frac{d}{dx} [\sqrt{2}] = 0 \].
04

Combine the results

Combine the results of differentiating each term: \[ f'(x) = 24x^3 - 21x^2 + 2 + 0 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

power rule
When working with derivatives, the power rule is one of the most fundamental tools. It's straightforward and applies to any term of the form \(x^n\). The power rule states that if you have a function \(f(x) = x^n\), then its derivative is \(f'(x) = nx^{n-1}\).
differentiation of constants
A critical concept in calculus is the differentiation of constants. When you differentiate a constant, it always results in zero. That's because a constant doesn't change, hence its rate of change (derivative) is zero. For example, if you have \(\frac{d}{dx} [\text{constant}] = 0\), no matter the value of the constant.
combining derivatives
After differentiating each term of a function separately, combining derivatives is the final step. This involves simply adding or subtracting the individual derivatives. For example, if you have \(f(x) = f_1(x) + f_2(x) + f_3(x)\), then \(f'(x) = f_1'(x) + f_2'(x) + f_3'(x)\). This allows us to handle even complex functions by breaking them down into simpler parts.

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Most popular questions from this chapter

After \(x\) weeks, the number of people using a new rapid transit system was approximately \(N(x)=6 x^{3}+500 x+8,000\). a. At what rate was the use of the system changing with respect to time after 8 weeks? b. By how much did the use of the system change during the eighth week?

s(t) denotes the position of an object moving along a line. $$ s(t)=\frac{2 t+1}{t^{2}+12} ; 0 \leq t \leq 4 $$

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Find the derivative of the given function. $$ f(x)=(3 x+1) \sqrt{6 x+5} $$
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