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Chapter 2: Problem 34

Find the derivative of the function. $$ h(x)=\frac{2 x^{2}-3 x+1}{x} $$

Short Answer

Expert verified
The derivative of the function \(h(x) = \frac{2x^{2}-3x+1}{x}\) is \(h'(x) = 2 - \frac{1}{x^{2}}\).

Step by step solution

01

Write Down the Function and Quotient Rule

The function is \( h(x)=\frac{2 x^{2}-3 x+1}{x} \). And the Quotient Rule states that the derivative of \(\frac{u}{v}\) is \(\frac{vu' - uv'}{v^{2}}\), where \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives.
02

Identify u, v, u' and v'

Here, \(u = 2x^2 - 3x + 1\), \(v = x\). So, \(u' = \frac{d}{dx}(2x^2 - 3x + 1) = 4x - 3\), \(v' = \frac{d}{dx}(x) = 1.\)
03

Apply the Quotient Rule

Substitute \(u\), \(v\), \(u'\), \(v'\) into the Quotient Rule to get: \(h'(x) = \frac{x(4x - 3) - (2x^{2} - 3x + 1)(1)}{x^{2}} = \frac{4x^{2} -3x -2x^2 +3x -1}{x^2} = \frac{2x^{2}-1}{x^2}.\)
04

Simplify the Result

Simplify the expression: \(h'(x) = \frac{2x^{2}-1}{x^2} = 2 - \frac{1}{x^{2}}\)

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Key Concepts

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

Quotient Rule
The quotient rule is an essential technique in calculus for finding the derivative of a quotient of two functions. If you have a function that can be expressed as the division of one function by another, denoted as \( \frac{u}{v} \), the quotient rule helps determine the derivative efficiently. This rule states:
  • The derivative of \( \frac{u}{v} \) is \( \frac{vu' - uv'}{v^2} \).
Here鈥檚 how it works:
  • Identify the functions: First, identify \( u \) and \( v \), where \( u \) is the numerator function and \( v \) is the denominator function.
  • Find the derivatives: Compute the derivatives \( u' \) and \( v' \).
  • Apply the rule: Plug these into the quotient rule formula to find the derivative.
Remember, this method helps manage complexity by breaking it into parts, making derivative calculations for quotients more organized and less error-prone.
Function Derivatives
Derivatives are fundamental in calculus, representing the rate of change of a function. For the exercise with \( h(x) = \frac{2x^2 - 3x + 1}{x} \), we begin by understanding that it's a quotient of two functions, \( u = 2x^2 - 3x + 1 \) and \( v = x \).
Calculating derivatives involves:
  • Finding individual derivatives: It's crucial first to differentiate \( u \, \) to get \( u' = 4x - 3 \), and \( v \), to get \( v' = 1 \), before applying the quotient rule.
  • Using notation correctly: Notations like \( \frac{d}{dx} \) signify differentiation with respect to \( x \), which is central to identifying derivatives of polynomial expressions like \( u \) and \( v \).
Understanding these basics ensures a solid foundation for tackling more complex differentiation problems.
Simplifying Expressions
After applying calculus rules such as the quotient rule, the resulting expression often requires simplification. Simplifying expressions is paramount because it provides a clearer and more comprehensible form of the derivative.
For \( h'(x) = \frac{2x^{2}-1}{x^2} \), simplification involved:
  • Combining like terms: In this solution, terms were simplified, such as combining \( 4x^2 \) and \( -2x^2 \).
  • Rewriting expressions: The goal is to break down the fraction to simpler terms: \( h'(x) = 2 - \frac{1}{x^2} \).
Simplification transforms complicated expressions into understandable and usable forms, essential for solving and interpreting calculus problems effectively.

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