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Chapter 1: Problem 17

Find the derivative of \(f(x)\) at the designated value of \(x\). f(x)=x^{3} \text { at } x=\frac{1}{2}

Short Answer

Expert verified
\(f'\left(\frac{1}{2}\right) = \frac{3}{4}\)

Step by step solution

01

- Recall the Derivative Rule

To find the derivative of a function, use the power rule. The power rule states that if you have a function of the form \(f(x) = x^n\), then the derivative, denoted as \(f'(x)\), is given by \(f'(x) = nx^{n-1}\).
02

- Apply the Power Rule

Given the function \(f(x) = x^3\), apply the power rule where \(n = 3\). Therefore, the derivative \(f'(x)\) is calculated as follows: \[ f'(x) = 3x^{3-1} = 3x^2\].
03

- Evaluate the Derivative at \(x = \frac{1}{2}\)

Now, substitute \(x = \frac{1}{2}\) into the derivative function: \[ f'\left(\frac{1}{2}\right) = 3\left(\frac{1}{2}\right)^2\].
04

- Simplify the Expression

Calculate the value: \[ \left(\frac{1}{2}\right)^2 = \frac{1}{4}\]. Therefore, \[ f'\left(\frac{1}{2}\right) = 3 \cdot \frac{1}{4} = \frac{3}{4}\].

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Key Concepts

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

Power Rule in Calculus
The power rule is a fundamental concept in calculus, especially when dealing with polynomial functions. This rule simplifies finding derivatives and is expressed as follows. For a function of the form \(f(x) = x^n\), the derivative is given by \(f'(x) = nx^{n-1}\). This rule is incredibly powerful because it allows you to differentiate polynomial terms quickly. Notice how the exponent \(n\) is brought down as a coefficient and then reduced by one. This consistency makes the power rule both easy to remember and apply. For example, given \(f(x) = x^3\), using the power rule results in \(f'(x) = 3x^2\).
Evaluating Derivatives
Once you've found the derivative using the power rule, the next step often involves evaluating this derivative at a specific point. Evaluation is simply substituting a given value of \x\ into the derivative function to find the rate of change at that point. For instance, if the derivative we found was \(f'(x) = 3x^2\), and we are asked to evaluate at \(x = \frac{1}{2}\), we proceed by substituting \(x = \frac{1}{2}\) into \(3x^2\). This gives us: \[f'\bigg(\frac{1}{2}\bigg) = 3\bigg(\frac{1}{2}\bigg)^2 = 3 \times \frac{1}{4} = \frac{3}{4}\bigg.\]. So, the derivative of the function at \(x = \frac{1}{2}\) is \frac{3}{4}\.
Basic Differentiation
Differentiation is at the heart of calculus and involves finding the derivative, which measures how a function changes as its input changes. The process starts by recognizing the type of function you're dealing with. Polynomial functions, like \(x^3\), are straightforward and directly applicable to the power rule. Here's the general differentiation process:
  • Identify the function form, such as \(x^n\).
  • Apply the power rule to find \(f'(x)\).
  • Evaluate the derivative at any specific point if required.
By following these steps, you can accurately and efficiently determine the rate at which a function's value is changing. Understanding these basics forms the foundation for approaching more complex calculus problems.

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Most popular questions from this chapter

Based on data from the U.S. Treasury Department, the federal debt (in trillions of dollars) for the years 1995 to 2004 was given approximately by the formula $$D(x)=4.95+.402 x-.1067 x^{2}+.0124 x^{3}-.00024 x^{4}$$ where \(x\) is the number of years elapsed since the end of \(1995 .\) Estimate the federal debt at the end of \(1999(x=4)\) and the rate at which it was increasing at that time.

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Find the indicated derivative. \(\frac{d}{d x}\left(x^{-1 / 3}\right)\)
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