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Chapter 3: Problem 56

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\frac{2 x}{7}+1$$

Short Answer

Expert verified
The derivative \( f'(x) = \frac{2}{7} \).

Step by step solution

01

Understand the Definition of a Derivative

The definition of the derivative of a function \( f(x) \) at a point \( x \) is given by the limit \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]. Our task is to apply this definition to find \( f'(x) \) for the function \( f(x) = \frac{2x}{7} + 1 \).
02

Substitute into the Derivative Definition

Substitute \( f(x) = \frac{2x}{7} + 1 \) and \( f(x+h) = \frac{2(x+h)}{7} + 1 \) into the derivative formula. The subtraction \( f(x+h) - f(x) \) becomes: \[ \frac{2(x+h)}{7} + 1 - \left(\frac{2x}{7} + 1\right) \].
03

Simplify the Expression

Simplify \( \frac{2(x+h)}{7} + 1 - \left(\frac{2x}{7} + 1\right) \). The 1s will cancel out, and the expression simplifies to: \[ \frac{2x + 2h}{7} - \frac{2x}{7} = \frac{2h}{7} \].
04

Calculate the Limit

Now, calculate the limit: \( \lim_{h \to 0} \frac{\frac{2h}{7}}{h} \). This simplifies to \( \lim_{h \to 0} \frac{2}{7} \), which is just \( \frac{2}{7} \) since it does not depend on \( h \).
05

State the Derivative

After evaluating the limit, we find that \( f'(x) = \frac{2}{7} \). This means that the derivative of the function \( f(x) = \frac{2x}{7} + 1 \) is a constant, \( \frac{2}{7} \).

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Key Concepts

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

Definition of Derivative
The derivative of a function is a fundamental concept in calculus. It essentially tells us how a function changes as its input changes. To find the derivative of a function, we use the definition:
  • The derivative of a function \( f(x) \) at a point \( x \) is expressed as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] This formula is crucial for calculating the rate of change of the function at any given point.
  • In our example, we apply this definition to the function \( f(x) = \frac{2x}{7} + 1 \).
Understanding this formula is key to mastering derivatives. It provides a systematic approach to finding the slope of the tangent line at any point on the function.
Limit Process
The limit process is an essential step in finding a derivative. It's the step where we see how the function behaves as its inputs change infinitesimally.
  • For our function \( f(x) = \frac{2x}{7} + 1 \), we substitute the expressions \( f(x) \) and \( f(x+h) \) into the derivative formula.
  • The subtraction \( f(x+h) - f(x) \) is simplified to \( \frac{2(x+h)}{7} + 1 - \left(\frac{2x}{7} + 1\right) \).
This results in the expression \( \frac{2h}{7} \) as the terms involving \( x \) and constant terms cancel out. The limit is then calculated as \( \lim_{h \to 0} \frac{\frac{2h}{7}}{h} \), which simplifies to \( \frac{2}{7} \). This shows how the difference in the function values becomes very small as \( h \) approaches zero.
Function Simplification
Function simplification is often necessary while working with derivatives to make calculations easier. This involves algebraically reducing complex expressions to simpler forms.
  • In this exercise, after substituting the expressions into the derivative formula, simplifying \( \frac{2(x+h)}{7} + 1 - \left(\frac{2x}{7} + 1\right) \) results in \( \frac{2h}{7} \).
  • This involves cancelling out the constant terms and terms involving \( x \).
Such simplifications not only make the expression easier to handle but also prepare it for applying the limit process. A clear and neat simplification ensures accuracy in calculating derivatives.
Constant Derivative
When calculating derivatives, sometimes you find a special case where the derivative is a constant value. This means that the rate of change of the function is uniform regardless of \( x \).
  • For the function \( f(x) = \frac{2x}{7} + 1 \), after completing the limit process, we found that \( f'(x) \) is \( \frac{2}{7} \).
  • This indicates a constant derivative, representing a straight-line slope for the linear function given.
A constant derivative tells us that the function's rate of change does not depend on \( x \), providing a straightforward understanding of linear functions in calculus. This makes constant derivatives a perfect example of uniform changes across an input range.

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Most popular questions from this chapter

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