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Chapter 29: Problem 3

Differentiate (a) \(y=6\) (b) \(y=6 x\).

Short Answer

Expert verified
(a) \(\frac{dy}{dx}=0\); (b) \(\frac{dy}{dx}=6\).

Step by step solution

01

Identify the type of function

For part (a), the function given is a constant, which is of the form \(y = c\), where \(c\) is a constant. For part (b), the function is linear, given in the form \(y = m x\), where \(m\) is the slope of the line.
02

Differentiate the Constant Function

The derivative of a constant function \(y = c\) is zero. This is because a constant function has no change as \(x\) changes. Therefore, for part (a), the derivative \(\frac{dy}{dx} = 0\).
03

Differentiate the Linear Function

For a linear function \(y = mx\), the derivative is simply \(m\) because the slope of the line is constant. Thus, for part (b), the derivative \(\frac{dy}{dx} = 6\).

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Key Concepts

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

Constant Function
A constant function is perhaps one of the simplest types of functions in mathematics. It is expressed in the form \( y = c \), where \( c \) is a constant value and does not depend on \( x \). This means that no matter what value \( x \) takes, \( y \) will always be the same. Hence, if you were to plot this on a graph, you would see a horizontal line.
  • It has no slope, as it doesn't rise or fall.
  • Its graph is a straight line parallel to the x-axis.
  • It is unchanging, and hence its derivative is always zero.
Understanding constant functions is crucial as it lays the groundwork for more complex functions. They exemplify a basic concept in calculus: a function representing no change.
Linear Function
A linear function is another fundamental concept in algebra and calculus. It is usually represented as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. For the purpose of differentiation, we simplify this to \( y = mx \), especially when \( b = 0 \).
  • The slope \( m \) determines how steep the line is.
  • If you graph a linear function, it results in a straight line.
  • Each unit increase in \( x \) leads to an increase of \( m \) units in \( y \).
This type of function shows a constant rate of change between variables, emphasizing why the slope \( m \) is key when finding the derivative.
Derivative
The derivative of a function is a fundamental tool in calculus that provides the rate of change or slope of the function at any given point. When we say we 'differentiate' a function, we mean we are finding its derivative.
  • It tells us how the function \( y \) changes with respect to \( x \).
  • For a constant function, the derivative is zero because a constant does not change.
  • For a linear function, the derivative equals the slope \( m \).
Understanding derivatives is crucial because it gives insights into how functions behave. It's like understanding the velocity of a moving car at any given second.
Calculus
Calculus is the branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided broadly into differential calculus and integral calculus.
  • Differential calculus focuses on finding derivatives and understanding slopes and rates of change.
  • Integral calculus focuses on the accumulation of quantities and areas under curves.
  • It helps in analyzing trends, predicting future outcomes, and optimizing functions.
Grasping the core concepts of calculus opens the doors to mastering the study of motion, change, and optimization, vital in science, engineering, and economics.

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