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Chapter 3: Problem 19

\(3-32\) Differentiate the function. \(F(x)=\left(\frac{1}{2} x\right)^{5}\)

Short Answer

Expert verified
The derivative of \( F(x) = \left( \frac{1}{2} x \right)^5 \) is \( F'(x) = \frac{5}{32} x^4 \).

Step by step solution

01

Identify the Function Type

The given function is \( F(x) = \left(\frac{1}{2} x\right)^{5} \). We recognize this as a power function of the form \( (ax)^n \), where \( a = \frac{1}{2} \) and \( n = 5 \). This requires the use of the power rule for differentiation.
02

Apply the Power Rule

The power rule for differentiation states that if \( y = (ax)^n \), then \( \frac{dy}{dx} = n(ax)^{n-1} \cdot a' \), where \( a' \) is the derivative of \( ax \) with respect to \( x \). Here, \( a' = a = \frac{1}{2} \). Therefore, the derivative is given by:\[ F'(x) = 5 \left( \frac{1}{2} x \right)^{5-1} \cdot \frac{1}{2} \]
03

Simplify the Derivative Expression

Calculate the expression: \( 5 \left( \frac{1}{2} x \right)^{4} \cdot \frac{1}{2} \). First, simplify \( \left( \frac{1}{2} x \right)^{4} = \left(\frac{1}{16} x^4\right) \). Then multiply by the constants:\[ F'(x) = 5 \times \frac{1}{16} x^4 \times \frac{1}{2} = \frac{5}{32} x^{4} \].
04

Write the Final Result

The derivative of the given function \( F(x) = \left( \frac{1}{2} x \right)^5 \) is \( F'(x) = \frac{5}{32} x^4 \). This is the simplified expression of the derivative.

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Key Concepts

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

Power Rule
The power rule is a quick way to find the derivative of a power function. This method is essential for students learning calculus, as it allows for the differentiation of functions like \((ax)^n\) effortlessly. The rule states that if you have a function \(y = (ax)^n\), the derivative \(\frac{dy}{dx}\) is calculated by bringing down the exponent \(n\) before the term, multiplying it by the derivative of the inner function, and reducing the exponent by one.
  • The formula is: \(\frac{dy}{dx} = n(ax)^{n-1} \cdot a'\), where \(a'\) is the derivative of \(ax\) with respect to \(x\).
  • For example, if \(a = \frac{1}{2}\) and \(n = 5\), your task becomes to compute \(5 \left( \frac{1}{2} x \right)^{4} \cdot \frac{1}{2}\).
The power rule simplifies the differentiation process significantly, making it a fundamental tool in calculus.
Power Function
A power function is an algebraic expression of the form \((ax)^n\), where \(a\) is a constant, \(x\) is the variable, and \(n\) is a real number exponent. Power functions form the building blocks of many mathematical expressions and are crucial for understanding more complex functions and equations in calculus.
  • Power functions often appear in many real-world scenarios, such as physics and engineering, due to their simple yet versatile structure.
  • When differentiating power functions, identifying the specific form \((ax)^n\) makes the application of the power rule straightforward, as seen in the exercise with the function \(F(x) = \left(\frac{1}{2} x\right)^{5}\).
Recognizing a power function is the first step in efficiently using the power rule for differentiation.
Derivative
The concept of a derivative is central to calculus. It represents the rate at which a function is changing at any given point and is often described as the function's slope. Differentiation, the process of finding a derivative, explains how the values of a function change concerning changes in \(x\).
  • The derivative \(F'(x)\) of a function \(F(x)\) is denoted using the expression \(\frac{dF}{dx}\), indicating the change in \(F\) with respect to \(x\).
  • Using the exercise example, the derivative \(F'(x) = \frac{5}{32} x^4\) tells us about the tangent slope of the function \(F(x)\) at any point \(x\).
Grasping the idea of derivatives helps in analyzing motion, growth, and other dynamics within various fields.
Calculus
Calculus is a branch of mathematics that deals with change and motion. It consists of two main areas: differential calculus, which focuses on finding derivatives, and integral calculus, which focuses on integrating functions. These areas help in understanding and modeling natural and technological phenomena.
  • In differential calculus, techniques like the power rule allow us to find the derivative of functions efficiently, as seen in the provided step-by-step solution.
  • Calculus is applicable in fields such as physics, economics, biology, and engineering. It enables the analysis of rates of change and the accumulation of quantities.
Learning calculus provides the tools to solve real-world problems involving dynamic systems, making it a crucial area of study in mathematics.

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Most popular questions from this chapter

(a) If \(\$ 3000\) is invested at 5\(\%\) interest, find the value of the investment at the end of 5 years if the interest is compounded ( i ) annually, (ii) semiannually, (iil) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b) If \(\mathrm{A}(\mathrm{t})\) is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by \(\mathrm{A}(\mathrm{t})\) .

A plane flying with a constant speed of 300 \(\mathrm{km} / \mathrm{h}\) passes over a ground radar station at an altitude of 1 \(\mathrm{km}\) and climbs at an angle of \(30^{\circ} .\) At what rate is the distance from the plane to the radar station increasing a minute later?

(a) If \(n\) is a positive integer, prove that $$\frac { d } { d x } \left( \sin ^ { n } x \cos n x \right) = n \sin ^ { n - 1 } x \cos ( n + 1 ) x$$ (b) Find a formula for the derivative of \(y = \cos ^ { n } x \cos n x\) that is similar to the one in part (a).

A lighthouse is located on a small island 3 \(\mathrm{km}\) away from the nearest point \(\mathrm{P}\) on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 \(\mathrm{km}\) from P?

Brain weight \(\mathrm{B}\) as a function of body weight \(\mathrm{W}\) in fish has been modeled by the power function \(\mathrm{B}=0.007 \mathrm{W}^{2 / 3}\) , where \(\mathrm{B}\) and \(\mathrm{W}\) are measured in grams. A model for body weight as a function of body length \(\mathrm{L}\) (measured in centimeters) is \(\mathrm{W}=0.12 \mathrm{L}^{2.53}\) . If, over 10 million years, the average length of a certain species of fish evolved from 15 \(\mathrm{cm}\) to 20 \(\mathrm{cm}\) at a constant rate, how fast was this species' brain growing when the average length was 18 \(\mathrm{cm} ?\)
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