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Chapter 1: Problem 41

Find \(f^{\prime}(x)\). $$f(x)=4 x-7$$

Short Answer

Expert verified
The derivative of \[ f(x) = 4x - 7 \] is \[ f^{\textprime}(x) = 4 \].

Step by step solution

01

Understand the function

The given function is a linear function in the form of \[ f(x) = 4x - 7 \].
02

Recall the derivative rules

For a linear function of the form \[ f(x) = ax + b \], the derivative is given by the coefficient of the variable \( x \). This means the derivative of \( ax \) is \( a \), and the derivative of a constant \( b \) is 0.
03

Apply the derivative rules

Using the rules, calculate the derivative of each term separately:- The derivative of the term \( 4x \) is \( 4 \).- The derivative of the constant \( -7 \) is \( 0 \).
04

Combine the results

Combine the results from Step 3 to find \( f^{\textprime}(x) \).Since the derivative of \( 4x \) is \( 4 \) and the derivative of \( -7 \) is \( 0 \), the final result is:\[ f^{\textprime}(x) = 4 \].

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Key Concepts

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

Linear Function
A linear function is a type of function that represents a straight line when graphed. It usually takes the form \[ f(x) = ax + b \], where:
  • **'a'** is the slope of the line or the rate at which y changes as x changes.
  • **'x'** is the independent variable.
  • **'b'** is the y-intercept or the point where the line crosses the y-axis.
In our example, the linear function given is \( f(x) = 4x - 7 \). Here, **'a'** is 4 and **'b'** is -7. It means that for every unit increase in **'x'**, **'f(x)'** increases by 4 units, and the line crosses the y-axis at **-7**. Linear functions are simple but powerful tools in modeling relationships where the rate of change is constant.
Derivative Rules
Derivatives are fundamental in calculus for understanding how a function changes. They measure the rate at which a quantity changes. For linear functions, some rules help us find the derivative quickly and easily. Here are two important rules:
  • The derivative of a term with **'x'** is the coefficient of **'x'**.
  • The derivative of a constant term (a number without **'x'**) is always zero.
Applying these rules, we can find the derivative of each term in our function \( f(x) = 4x - 7 \):
  • The derivative of **4x** is **4** because **4** is the coefficient of **'x'**.
  • The derivative of **-7** is **0** because **-7** is a constant.
Combining these, we get \( f^{\textprime}(x) = 4 + 0 = 4 \).
Constant Term
A constant term in a function is a number on its own. It does not change because it is not multiplied by a variable like **'x'**. In our example, \( f(x) = 4x - 7 \), **-7** is the constant term. When finding the derivative of a linear function, it's important to remember that the derivative of any constant term is zero. This is because a constant term does not change; its rate of change is zero. Regardless of the value of **x**, the constant term remains the same.
In summary:
  • Identifying the constant helps in calculating the derivative correctly.
  • Ignoring the constant term while differentiating will lead to an incorrect result.
Therefore, always remember: derivative of any constant term equals zero!

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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. $$y=(\sqrt{x}+\sqrt[3]{x})^{2}$$

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=\frac{5 x^{2}+8 x-3}{3 x^{2}+2}$$

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=20 x^{3}-3 x^{5}$$

Use the Chain Rule to differentiate each function. You may need to apply the rule more than once. $$f(x)=\left(2 x^{5}+(4 x-5)^{2}\right)^{6}$$

For each function, find the interval(s) for which \(f^{\prime}(x)\) is positive. $$f(x)=\frac{1}{3} x^{3}-x^{2}-3 x+5$$
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