/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 \(3-32\) Differentiate the funct... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 3

\(3-32\) Differentiate the function. \(f(x)=186.5\)

Short Answer

Expert verified
The derivative of the function is 0.

Step by step solution

01

Understand the Function

The given function is a constant function, where \[f(x) = 186.5\]which means the function's value is 186.5 for all values of \(x\).
02

Apply Derivative of a Constant Rule

The derivative of a constant function \(f(x) = c\) is zero. Mathematically, this is expressed as:\[\frac{d}{dx} [c] = 0\]Thus, the derivative of \(f(x) = 186.5\) is zero.
03

Write the Derivative

The derivative of the given function can be written as:\[f'(x) = 0\]This expresses that the rate of change of a constant function with respect to \(x\) is zero.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Derivative of a Constant
When it comes to understanding the derivative of a constant, it is important to know that constant functions have no variables. A constant function, like the one we're looking at \(f(x) = 186.5\), maintains the same value no matter what input you provide.
The key idea here is that because a constant doesn't change, its rate of change is zero. In calculus, this is a core concept. Mathematically, the derivative of a constant \(c\) is always zero:
  • Write the constant as \(f(x) = c\).
  • Take the derivative: \(\frac{d}{dx}[c] = 0\).
This simplicity is what makes constant function differentiation more approachable for beginners.
Rate of Change
In calculus, the derivative of a function represents its rate of change. It shows how much the function's value changes as the input changes. For example, if you have a slope of a line, the derivative tells how steep the line is.
For constant functions like \(f(x) = 186.5\), there is no slope. The line is perfectly horizontal. This is why the derivative, or rate of change, is zero.
  • No matter the x-value, the output, \(f(x)\), remains the same.
  • Because there's no increase or decrease in value, the rate of change remains zero.
Understanding this simplifies the concept of derivatives for constant functions.
Differentiation Rules
Differentiation rules are a fundamental part of calculus. They provide a shorthand for finding derivatives quickly without going back to the limit definitions every time.
For constant functions, there's a straightforward rule: the derivative of a constant is zero. This fits into a much larger toolbox of differentiation rules. Other common rules include:
  • The power rule: for functions like \(x^n\), the derivative is \(nx^{n-1}\).
  • The product rule: used when multiplying two functions, \[ \(f(x)g(x)\) has the derivative: \(f'(x)g(x) + f(x)g'(x)\). \]
  • The quotient rule: for dividing one function by another, \[ \(\frac{f(x)}{g(x)}\) becomes \(\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\). \]
These rules help in breaking down complex functions into manageable parts to find their derivatives more easily.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A runner sprints around a circular track of radius 100 \(\mathrm{m}\) at a constant speed of 7 \(\mathrm{m} / \mathrm{s}\) . The runner's friend is standing at a distance 200 \(\mathrm{m}\) from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 \(\mathrm{m} ?\)

Show that if \(a \neq 0\) and \(b \neq 0,\) then there exist numbers \(\alpha\) and \(\beta\) such that \(a e^{x}+b e^{-x}\) equals either \(\alpha \sinh (x+\beta)\) or \(\alpha \cosh (x+\beta) .\) In other words, almost every function of the form \(f(x)=a e^{x}+b e^{-x}\) is a shifted and stretched hyperbolic sine or cosine function.

On page 431 of Physics: Calculus, 2\(d\) ed, by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, \(2000 ),\) in the course of deriving the formula \(\mathrm{T}=2 \pi \sqrt{\mathrm{L} / \mathrm{g}}\) for the period of a $$ \begin{array}{l}{\text { pendulum of length } L, \text { the author obtains the equation }} \\ {\mathrm{a}_{\mathrm{T}}=-g \sin \theta \text { for the tangential acceleration of the bob of the }}\end{array} $$ pendulum, He then says, "for small angles, the value of \(\theta\) in radians is very nearly the value of \(\sin \theta\) ; they differ by less than 2\(\%\) out to about \(20^{\circ} .\) $$ \begin{array}{c}{\text { (a) Verify the linear approximation at } 0 \text { for the sine function: }} \\ {\sin x=x}\end{array} $$ (b) Use a graphing device to determine the values of \(x\) for which sin \(x\) and \(x\) differ by less than 2\(\%\) . Then verify Hecht's statement by converting from radians to degrees.

The edge of a cube was found to be 30 \(\mathrm{cm}\) with a possible error in measurement of 0.1 \(\mathrm{cm}\) . Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube.

If \(R\) denotes the reaction of the body to some stimulus of strength \(x,\) the sensitivity \(S\) is defined to be the rate of change of the reaction with respect to \(x\) . A particular example is that when the brightness \(x\) of a light source is increased, the eye reacts by decreasing the area \(R\) of the pupil. The experimental formula $$R=\frac{40+24 x^{0.4}}{1+4 x^{0.4}}$$ has been used to model the dependence of \(\mathrm{R}\) on \(\mathrm{x}\) when \(\mathrm{R}\) is measured in square millimeters and \(\mathrm{x}\) is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both \(\mathrm{R}\) at low levels of of \(\mathrm{x} .\) Comment on the values of \(\mathrm{R}\) and \(\mathrm{S}\) at low levels of brightness. Is this what you would expect?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.