/*! 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 38 Find \(f^{\prime}(x)\) by using ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 38

Find \(f^{\prime}(x)\) by using the definition of the derivative. $$f(x)=x^{4}$$$

Short Answer

Expert verified
The derivative \( f'(x) = 4x^3 \).

Step by step solution

01

Write the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined as \( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \). This formula will be used to find the derivative of \( f(x) = x^4 \).
02

Substitute \( f(x) = x^4 \) into the Definition

Substitute the function \( f(x) = x^4 \) into the derivative definition: \[ f'(x) = \lim_{{h \to 0}} \frac{{(x+h)^4 - x^4}}{h} \].
03

Expand \((x+h)^4\)

Expand the expression \((x+h)^4\) using the binomial theorem: \((x+h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).
04

Simplify the Numerator

Substitute the expanded form back into the difference quotient: \[ f'(x) = \lim_{{h \to 0}} \frac{{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}}{h} \].Cancel out \( x^4 \): \[ f'(x) = \lim_{{h \to 0}} \frac{{4x^3h + 6x^2h^2 + 4xh^3 + h^4}}{h} \].
05

Factor Out \(h\) from the Numerator

Factor \(h\) out of the numerator: \[ f'(x) = \lim_{{h \to 0}} \frac{h(4x^3 + 6x^2h + 4xh^2 + h^3)}{h} \].Cancel \(h\) from the numerator and denominator: \[ f'(x) = \lim_{{h \to 0}} (4x^3 + 6x^2h + 4xh^2 + h^3) \].
06

Evaluate the Limit as \(h \to 0\)

As \(h\) approaches zero, the terms containing \(h\) vanish, resulting in: \[ f'(x) = 4x^3 \].

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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, representing the rate at which a function changes at any given point. The formal definition involves a limit,
  • Let's consider a function, denoted as \( f(x) \).
  • The derivative, \( f'(x) \), is given by this limit expression: \( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \).
This formula captures the idea of calculating the "instantaneous rate of change" or the "slope" of the function at a specific point. To simplify, think of it as examining the change in the function's output as you make an incredibly tiny change in its input (represented by \( h \)). By making \( h \) approach zero, we precisely determine this rate without interruption from other variables.
Binomial Theorem
When dealing with powers of binomials, such as \((x+h)^n\), the binomial theorem becomes a useful tool. This theorem provides a formula to expand expressions of the form \((a + b)^n\), breaking them down into a series of terms. For example,
  • With \((x+h)^4\), it expands into \( x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).
Each term in the expansion follows a pattern:
  • Coefficients come from the binomial coefficients, which form the "Pascal's Triangle." The coefficients for \((x+h)^4\) are 1, 4, 6, 4, 1.
  • The powers decrease for \( x \) and increase for \( h \), providing a smooth algebraic expansion to work with.
With this, one can easily manage expressions during the differentiation process, just like in the original exercise.
Limit Evaluation
Limits are a cornerstone of calculus and a critical part of finding derivatives. They allow us to handle values that approach a certain point. When evaluating limits during differentiation, especially with expressions like the difference quotient,
  • You simplify expressions by canceling common terms, as was done with \( x^4 \) in the example.
  • Factor out variables such as \( h \), enabling you to handle expressions more easily.
  • As \( h \) approaches zero, any term still involving \( h \) vanishes, leaving only the terms independent of \( h \).
This step-by-step simplification and limit evaluation help you pinpoint the exact derivative of a function accurately.
Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers, often combined with coefficients. Understanding how derivatives apply to them is crucial. In our specific case,
  • The function is a simple polynomial, \( f(x) = x^4 \).
  • When finding its derivative, the terms align with specific rules of calculus, particularly involving power functions.
  • You calculate the derivative: any term \( ax^n \) results in \( anx^{n-1} \).
Polynomials have straightforward derivatives due to this pattern, making them a good starting point when learning calculus. The resulting expression after differentiation reveals a new function, indicating how the original polynomial modifies at every point.

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Most popular questions from this chapter

The carbon monoxide level in a city is predicted to be \(0.02 x^{3 / 2}+1\) ppm (parts per million), where \(x\) is the population in thousands. In \(t\) years the population of the city is predicted to be \(x(t)=12+2 t\) thousand people. Therefore, in \(t\) years the carbon monoxide level will be $$ P(t)=0.02(12+2 t)^{3 / 2}+1 \quad \overline{\mathrm{ppm}} $$ Find \(P^{\prime}(2)\), the rate at which carbon monoxide pollution will be increasing in 2 years.

Status A study estimated how a person's social status (rated on a scale where 100 indicates the status of a college graduate) depended on years of education. Basedon this study, with \(e\) years of education, a person's status is \(S(e)=0.22(e+4)^{21}\). Find \(S^{\prime}(12)\) and interpret your answer.

The windchill index (revised in 2001 ) for a temperature of 32 degrees Fahrenheit and wind speed \(x\) miles per hour is \(W(x)=55.628-22.07 x^{0.16}\). a. Graph the windchill index on a graphing calculator using the window \([0,50]\) by \([0,40]\). Then find the windchill index for wind speeds of \(x=15\) and \(x=30\) mph. b. Notice from your graph that the windchill index has first derivative negative and second derivative positive. What does this mean about how successive 1-mph increases in wind speed affect the windchill index? c. Verify your answer to part (b) by defining \(1 / 2\) to be the derivative of \(y_{1}\) (using NDERIV), evaluating it at \(x=15\) and \(x=30\), and interpreting your answers.

The population of a city \(x\) years from now is predicted to be \(P(x)=\sqrt[4]{x^{2}+1}\) million people for \(1 \leq x \leq 5 .\) Find when the population will be growing at the rate of a quarter of a million people per year. [Hint: On a graphing calculator, enter the given population function in \(y_{1}\), use NDERIV to define \(1 / 2\) to be the derivative of \(y_{1}\), and graph both on the window \([1,5]\) by \([0,3]\). Then TRACE along \(y_{2}\) to find the \(x\) -coordinate (rounded to the nearest tenth of a unit) at which the \(y\) -coordinate is \(0.25\). You may have to \(Z O O M\) IN to find the correct \(x\) -value.]

Use the Generalized Power Rule to find the derivative of each function. $$y=\left(\frac{1}{w^{4}+1}\right)^{5}$$
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