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Chapter 2: Problem 22

In Exercises 3鈥�24, use the rules of differentiation to find the derivative of the function. $$ y=7+\sin x $$

Short Answer

Expert verified
The derivative of the function \(y = 7 + \sin x\) is \(\cos x\).

Step by step solution

01

Identify Functions

The given function is \(y = 7 + \sin x\) where \(7\) is a constant and \(\sin x\) is a trigonometric function.
02

Derive the constant term

The derivative of a constant is always zero. Hence the derivative of \(7\) will be \(0\).
03

Derive the trigonometric function

The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\). So the derivative of the given trigonometric function will be \(\cos x\).
04

Add the derivatives together

Since our function is an addition of a constant and a function, the derivative of the function \(y = 7 + \sin x\) will be a summation of their derivatives. In other words, it will be \(0 + \cos x\) or simply \(\cos x\).

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are fundamental in mathematics and have a variety of applications in science and engineering. They originate from the study of triangles, especially right triangles. The basic trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each function represents a ratio of sides in a right triangle, and they are periodic, which means they repeat values in regular intervals. This periodicity is crucial for modeling waves and oscillations.
Trigonometric functions can be used to describe patterns and relationships in both two-dimensional and three-dimensional space.
  • The sine function, \( \sin x \), oscillates between -1 and 1 and is one of the key functions to learn in trigonometry.
  • These functions are also extended to complex numbers, finding uses in signal processing and acoustics.
Derivative of Sine Function
The derivative of a function measures how the function value changes as its input changes. For trigonometric functions like sine, this involves figuring out how steep or flat the slope is at any point on the sine curve. This is incredibly useful in physics for computing velocities and in engineering for design.
For the sine function, \( \sin x \), taking the derivative involves applying basic rules of differentiation. The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). This result makes intuitive sense because \( \cos x \) is the slope of the sine curve at each point.
  • This differentiation rule is foundational and is often one of the first examples students encounter when learning calculus.
  • The relationship between sine and cosine in differentiation also illustrates the notable waves and oscillations. Understanding this will aid in grasping concepts involving harmonic motion and circuits.
Constant Function
A constant function is one of the simplest types of functions you can encounter. It can be expressed as \( f(x) = c \), where \( c \) is a fixed number. Graphically, it is represented as a flat, horizontal line in the coordinate plane.
One of the most important aspects of a constant function is that its derivative is zero. This means that no matter the value of \( x \), the rate of change is always zero. This concept is often encountered in introductory calculus when learning the rules of differentiation.
  • In practical terms, no change in the function's value means no slope. The function is consistent everywhere鈥攐ffering a stark contrast to functions like sine which oscillate.
  • Pondering about constant functions with no change offers intuitive insights that bolster understanding of calculus fundamentals.

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Most popular questions from this chapter

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\tan x ; \frac{d x}{d t}=3 \text { feet per second }} \\\ {\begin{array}{llll}{\text { (a) } x=-\frac{\pi}{3}} & {\text { (b) } x=-\frac{\pi}{4}} & {\text { (c) } x=0}\end{array}}\end{array} $$

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\cos x ; \frac{d x}{d t}=4 \text { centimeters per second }} \\ {\begin{array}{llll}{\text { (a) } x=\frac{\pi}{6}} & {\text { (b) } x=\frac{\pi}{4}} & {\text { (c) } x=\frac{\pi}{3}}\end{array}}\end{array} $$

True or False? In Exercises \(125-128\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=\sin ^{2}(2 x),\) then \(f^{\prime}(x)=2(\sin 2 x)(\cos 2 x)\).

True or False? In Exercises \(125-128\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y\) is a differentiable function of \(u,\) and \(u\) is a differentiable function of \(x,\) then \(y\) is a differentiable function of \(x .\)

Acceleration The velocity of an object in meters per second is $$ v(t)=36-t^{2} $$ for \(0 \leq t \leq 6 .\) Find the velocity and acceleration of the object when \(t=3 .\) What can be said about the speed of the object when the velocity and acceleration have opposite signs?
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