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Chapter 1: Problem 42

Find \(f^{\prime}(x)\). $$f(x)=7 x-14$$

Short Answer

Expert verified
The derivative is \( f^{\text{'} }(x) = 7 \).

Step by step solution

01

Identify the Function

First, recognize the given function: \[ f(x) = 7x - 14 \].
02

Understand the Basic Derivative Rule

Recall that the derivative of a linear function, \( ax + b \), is simply \( a \).
03

Differentiate the Function

Using the basic rule of derivatives, differentiate each term of the function:- The derivative of \(7x\) is \(7\), because the derivative of \(ax\) is \(a\).- The derivative of \(-14\) is \(0\), because the derivative of any constant is \(0\).
04

Combine Results

Add the results of the differentials: \[ f^{\text{'} }(x) = 7 + 0 = 7 \].

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Key Concepts

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

Basic Derivative Rules
Let's start with understanding some basic derivative rules. Derivatives are a fundamental part of calculus, allowing us to find the rate at which a function changes. When we have a linear function like f(x) = 7x - 14 , these rules make it straightforward to find its derivative.
The most important rules to remember for linear functions are:
  • The derivative of a constant is zero.
  • The derivative of a term like ax is just the coefficient a .
So, when differentiating polynomials, you simply apply these rules term-by-term. Let's see how this applies to our given function f(x) = 7x - 14 .
Differentiation
Differentiation is the process of finding the derivative of a function. It shows us the rate of change of the function's output concerning its input. For the function f(x) = 7x - 14 , let's break down the differentiation process step-by-step.
First, identify and separate the terms in the function: 7x and -14 . Using our basic derivative rules:
  • The derivative of 7x is 7 , because the coefficient of x is 7 .
  • The derivative of -14 is 0 , as it's a constant.
Now, sum the derivatives of each term:
f'(x) = 7 + 0 = 7
Therefore, the derivative of f(x) = 7x - 14 , or f'(x) , is simply 7 .
Calculus
Calculus is the branch of mathematics that deals with continuous change. It is divided into two main parts: differential calculus and integral calculus. Here, we focus on differential calculus, which involves finding derivatives.
The derivative of a function gives us a precise way to measure how that function changes. Let's relate this to our provided problem: f(x) = 7x - 14 . The solution process embodies fundamental calculus concepts:
  • Recognition of the function type (linear).
  • Application of basic derivative rules.
  • Execution of differentiation to find the rate of change (derivative).
Understanding these concepts helps build a strong foundation in calculus, empowering you to tackle more complex problems effectively. The linear nature of the function made the differentiation straightforward, and we found that f'(x) = 7 .

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Most popular questions from this chapter

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

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