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Chapter 1: Problem 21

Find the derivative of \(f(x)\) at the designated value of \(x\). \(f(x)=x+11\) at \(x=0\)

Short Answer

Expert verified
The derivative of \( f(x) = x + 11 \) at \( x = 0 \) is 1.

Step by step solution

01

Identify the Function

The given function is \( f(x) = x + 11 \).
02

Determine the Form of the Derivative

The derivative of a linear function \( f(x) = mx + b \) is \( f'(x) = m \). Here, the function \( f(x) = x + 11 \) is linear with a slope of 1.
03

Differentiate the Function

Differentiate \( f(x) = x + 11 \). Using the basic differentiation rules, the derivative \( f'(x) \) is 1.
04

Evaluate the Derivative at \(x = 0\)

Substitute \( x = 0 \) into the derivative to find \( f'(0) \). Therefore, \( f'(0) = 1 \).

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Key Concepts

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

Derivative
A derivative represents the rate of change of a function with respect to a variable. It's like the slope of a curve at any given point. In simpler terms, if you have a function that tells you the position of something over time, the derivative of that function tells you the speed. In mathematics, the derivative is often denoted as \( f'(x) \) or \( \frac{dy}{dx} \). For our function \( f(x) = x + 11 \), the derivative will tell us how it changes as \( x \) changes. Since this is a simple function, the rate of change is constant.
Linear Function
A linear function is a function that creates a straight line when graphed. This means that it has a constant rate of change, or slope. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our example, \( f(x) = x + 11 \), the slope \( m \) is 1 and the y-intercept \( b \) is 11. Slope 1 means that for each one unit increase in \( x \), the function \( f(x) \) also increases by one unit. The y-intercept 11 means that when \( x = 0 \), the value of the function \( f(x) \) is 11.
Basic Differentiation Rules
To find the derivative of a function, you often use some basic rules. The most common rules are:
  • Constant Rule: The derivative of a constant is zero. For example, \( \frac{d}{dx}[c] = 0 \).
  • Power Rule: The derivative of \( x^n \) is \( nx^{n-1} \). For example, \( \frac{d}{dx}[x^2] = 2x \).
  • Sum Rule: The derivative of a sum of functions is the sum of their derivatives. For example, \( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \).
  • Linearity Rule: The derivative of a linear function \( f(x) = mx + b \) is \( m \). In our exercise, \( f(x) = x + 11 \) fits this rule with \( m = 1 \).
Evaluate Derivative
To evaluate a derivative, you simply follow these steps:
  • Find the derivative of the function.
  • Substitute the specific value of \( x \) into the derivative function.
For the given exercise, let's evaluate the derivative at \( x = 0 \):
  • We have the function \( f(x) = x + 11 \).
  • The derivative is obtained as \( f'(x) = 1 \) according to the Linearity Rule.
  • Now, substitute \( x = 0 \) into \( f'(x) \): Thus, \( f'(0) = 1 \).
The result tells us that for the given linear function \( f(x) = x + 11 \), the rate at which the function changes at \( x = 0 \) is 1.

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Most popular questions from this chapter

Use paper and pencil to find the equation of the tangent line to the graph of the function at the designated point. Then, graph both the function and the line to confirm it is indeed the sought-after tangent line. \(f(x)=x^{3},(1,1)\)

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