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Chapter 2: Problem 43

Find the derivative of the function. \(f(x)=\frac{x^{3}-3 x^{2}+4}{x^{2}}\)

Short Answer

Expert verified
The derivative of the function \(f(x)=\frac{x^{3}-3 x^{2}+4}{x^{2}}\) is \(f'(x)=\frac{x^{4}-8x}{x^{4}}.\)

Step by step solution

01

Identify and Take Derivatives of the Numerator and Denominator

Identify the numerator \(u=x^{3}-3 x^{2}+4\) and the denominator \(v=x^{2}\). Take their derivatives: \(u'=3x^{2}-6x, v'=2x\)
02

Apply the Quotient Rule for Derivatives

Apply the quotient rule \(\frac{d}{dx}[\frac{u}{v}] = [\frac{vu'-uv'}{v^2}]\). Substituting \(u', u, v', v\) values gives \(\frac{(x^{2})(3x^{2}-6x)-(x^{3}-3x^{2}+4)(2x)}{(x^{2})^{2}}\) = \(\frac{3x^{4}-6x^{3}-2x^{4}+6x^{3}-8x}{x^{4}}\)
03

Simplify the Expression

Simplify the above expression to obtain the derivative of the given rational function: \(f'(x)=\frac{x^{4}-8x}{x^{4}}\)
04

Simplify Further if Possible

After checking, the resultant expression cannot be simplified further.

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Key Concepts

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

Quotient Rule for Derivatives
When dealing with rational functions, the quotient rule is your best friend for finding derivatives.
If you have a function that is a ratio of two functions, say\( \frac{u(x)}{v(x)} \), you can apply the quotient rule to find its derivative efficiently.
The quotient rule is defined as:
  • \( \frac{d}{dx}\left[ \frac{u}{v}\right] = \frac{v \cdot u' - u \cdot v'}{v^2} \)
  • Where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives.
To apply the quotient rule:
  • Identify the numerator \( u \) and its derivative \( u' \).
  • Identify the denominator \( v \) and its derivative \( v' \).
  • Plug these into the formula and simplify the expression.
Practice this process step-by-step, and it will become intuitive!
Simplifying Expressions After Differentiation
After applying the quotient rule, you'll often find yourself with a complex expression. This expression deserves some attention to make it cleaner and maybe even simpler.
The key steps for simplifying include:
  • Combine like terms. This is where terms with the same power of \( x \) come together.
  • Factor out any common terms if possible. This can really cut down on the length and complexity of your expression.
  • Simplify fractions by canceling common factors in the numerator and denominator, if applicable.
In our original solution, after the initial application of the quotient rule, we simplified the complex fraction to \( \frac{x^{4} - 8x}{x^{4}} \).
However, there were no further factors common between the numerator and denominator, so this was as simple as it got.
Remember, every simplification step helps in highlighting the main features of the expression.
Understanding Rational Function Derivatives
Derivatives of rational functions are fundamental, as they appear frequently across different areas of calculus and their applications.
Rational functions are simply ratios of polynomial functions, for example, \( f(x) = \frac{x^3 - 3x^2 + 4}{x^2} \).
When you differentiate a rational function, you're looking for how the rate of change of these ratios behaves.
Quotient rule is the main tool you use, as it efficiently handles the division between two functions.
  • First and foremost, ensure you're comfortable identifying parts of the rational function as numerator \( u \) and denominator \( v \).
  • Next, understand the role of each derivative \( u' \) and \( v' \) in calculating the overall derivative.
  • Lastly, remember simplifying your results often helps in understanding the function's characteristics better.
By practicing these steps, you'll grow adept at handling any rational function derivative you encounter.

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Most popular questions from this chapter

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