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Chapter 3: Problem 36

Find the first and second derivatives of the functions in Exercises 33-40. \(u=\frac{\left(x^{2}+x\right)\left(x^{2}-x+1\right)}{x^{4}}\)

Short Answer

Expert verified
First derivative: \(-\frac{3}{x^4}\); Second derivative: \(\frac{12}{x^5}\).

Step by step solution

01

Understand the Function

The function given is a rational function: \( u = \frac{(x^2 + x)(x^2 - x + 1)}{x^4} \). This means we have a numerator \( (x^2 + x)(x^2 - x + 1) \) and a denominator \( x^4 \). The first task is to simplify the expression if possible.
02

Simplify the Function

Expand the numerator: \( (x^2 + x)(x^2 - x + 1) = x^4 - x^3 + x^2 + x^3 - x^2 + x = x^4 + x \). After simplifying, the function becomes \( u = \frac{x^4 + x}{x^4} = 1 + \frac{1}{x^3} \). This is a simpler form that will make differentiation easier.
03

Find the First Derivative

Using the power rule and the derivative of constants, differentiate \( u = 1 + x^{-3} \). The derivative of 1 is 0, and the derivative of \( x^{-3} \) is \( -3x^{-4} \). Therefore, \( u' = 0 - 3x^{-4} = -3x^{-4} \).
04

Find the Second Derivative

Differentiate \( u' = -3x^{-4} \) again to find the second derivative. Using the power rule, the derivative of \( -3x^{-4} \) is \( 12x^{-5} \). Thus, the second derivative is \( u'' = 12x^{-5} \).
05

Write the Derivatives in a More General Form

If desired, express the derivatives in terms of positive exponents. First derivative: \( u' = -\frac{3}{x^4} \). Second derivative: \( u'' = \frac{12}{x^5} \). This form is often used to make it easier to evaluate functions at specific points.

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Key Concepts

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

Rational Function
Rational functions are a type of mathematical function expressed as the ratio of two polynomials. In the given exercise, the function is characterized as:
  • The numerator: a polynomial formed by multiplying \(x^2 + x\) and \(x^2 - x + 1\).
  • The denominator: \(x^4\), which is a simpler polynomial.
To work with rational functions effectively, simplifying them is often beneficial. In this case, expanding the numerator and simplifying it makes the arithmetic less complex. This not only clarifies the expression but also aids in differentiation. A simplified rational function will allow for easier application of derivative rules.
Power Rule
The power rule is a key tool in calculus for differentiating functions that are powers of \(x\). The general principle states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
Knowing this simplifies the computation of derivatives.
Applying the power rule in this exercise:
  • For \(x^{-3}\), the derivative becomes \( -3x^{-4} \).
  • Proceeding from this, differentiating \( -3x^{-4} \) yields \( 12x^{-5} \).
Using the power rule effectively transforms cumbersome expressions into manageable derivatives. The coefficients fall naturally into place, streamlining the process. This rule is essential in calculus for quickly finding derivatives of polynomial terms.
First Derivative
The first derivative of a function provides information about the rate of change of the function with respect to \(x\). It is essentially the slope of the tangent line at any point on the curve of the function.
For the given problem, the simplified function \(u = 1 + x^{-3}\) leads to the first derivative:
  • Calculate \(u' = 0 - 3x^{-4}\) using basic differentiation rules.
  • Express it as \(-\frac{3}{x^4}\) with a positive exponent for clearer interpretation.
This derivative shows how the function's rate of change varies with \(x\), getting steeper or shallower depending on the context. It's a foundation for understanding the function's behavior over its domain.
Second Derivative
The second derivative provides insights into the curvature or concavity of a function's graph. It indicates the rate of change of the first derivative.
In this exercise, finding the second derivative involves:
  • Differentiating the first derivative \(u' = -3x^{-4}\), giving \(u'' = 12x^{-5}\).
  • Expressing \(u'' = \frac{12}{x^5}\) using positive exponents.
This second derivative tells us whether the function is curving upwards or downwards. It helps understand the nature and behavior of the function's graph at different points. Positive values indicate upward concavity, while negative ones point to downward concavity, offering insights into potential inflection points.

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